Polynomial Identities of Finite Dimensional Lie Algebras

A.V.Iltyakov

Author address: School of Mathematics and Statistics, the University of Sydney, NSW,

2006, Australia

E-mail address: aleks@maths.usyd.edu.au

Contents

Preface 4

Chapter 1. Kemer’s theorem 7 1. Algebras of generic elements 7 2. Linearization 8 3. Specializations of noncommutative polynomials 10 4. Decomposition of ideals of identities 15 5. Specht’s conjecture 17 6. Finite-dimensional embedding 19 7. Representability of relatively free algebras 21

Chapter 2. Identities of Lie algebra representations 25 1. Theorem 25 2. Capelli polynomials 26 3. Generalized Lie modules 28 4. Canonical decomposition 34 5. Cayley-Hamilton polynomial 39 6. Identities of irreducible representations 42 7. ”Counter-example” 45 8. Type of a ”counter-example” 47

Chapter 3. Finite basis of identities in finite dimensional Lie algebras 53 1. Linearization of a “counter-example” 53 2. Irreducible “counter-example” 58 3. Finite dimensional embedding 64 4. Canonical splitting 73 5. Main Lemma 76 6. Sorting of variables 80 7. Final step 86

Bibliography 93

Preface

During a long time one of the central problem in the theory of associative algebras satisfying a polynomial identity (PI algebras) was Specht’s conjecture [36] saying that any associative algebra over a field of characteristic 0 has a finite basis of polynomial identities. A positive solution was given by A.R.Kemer in 1987, see [19]. His proof is based on a geometric approach due to Amitsur [2] and accumulates ideas and results of many specialists on PI rings and algebras, see, e.g., [27, 6, 33, 34].

The underlying idea is that free generators of a relatively free algebra are regarded as independent (vector) variables taking value in an algebra A (generic elements of A), so noncommutative polynomials are contained in the algebra of mappings in many varibales from A to A with pointwise operations. In fact, A.Kemer proved a more general statement saying, in particular, that a relatively free PI algebra of finite rank k is canonically isomorphic to the algebra of k generic elements of a finite dimensional algebra. It is a far-reaching generalization of a basic fact that over an infinite field the algebra of usual polynomials in k variables is isomorphic to the algebra of regular functions of the k-dimensional affine space.

The success in the associative case stimulated attempts to get a similar statement in varieties of nonassociative algebras. Thus, A.Ya.Vais and E.I.Zel’manov [38] proved an analog of Kemer’s theorem on finite basis of polynomial identities for finitely generated Jordan PI algebras. In the class of Lie algebras the first example of a finite dimensional algebra which has no a finite basis of identities was constructed by M.Vaughan-Lee [39]; it was essential that the ground field was infinite and of characteristic 2. Later V.Drensky generalized it to the case of an infinite field of arbitrary positive characteristic [12]. On the other hand, by a theorem due to Ju.A.Bakhturin and A.Ju.Ol’shanskii [4], any finite dimensional Lie algebra over a finite field has a finite basis of identities; that was an analog of a well-known theorem in Group Theory proven by S.Oates and M.B.Powell [28, 26]. So, the case of characteristic zero was especially interesting, see [10, p.43], [33], [14], [6, p.188]; some important particular results were obtained in [33, 22].

Using Kemer’s approach, the author gave an affirmative answer to this conjecture [16]. Moreover, the way used in that work forced to prove a similar statement in a more general context, in the class of PI representations of finitely generated Lie algebras. The aim of this notes is to give the proof of that result in a revised form. For convenience of the reader we also include Kemer’s theorem about representability of an associative relatively free algebra of finite rank in terms of generic elements of a finite dimensional algebra; in the main, the proof is similar to the original, however we have changed notations and some technical arguments to prepare the reader to the Lie algebra case.

Although the analog of Kemer’s theorem about finite basis of identities of a finite dimensional Lie algebra takes place, it is still unknown whether the relatively free Lie algebra of finite rank, whose multiplication algebra satisfies a polynomial identity, is the algebra of generic elements of a finite dimensional Lie algebra. A similar question is still open in varieties of alternative and Jordan algebras. It would be also interesting to outline the class of algebras where the problem of finite basis of identities in the finite dimensional case can be resolved in a similar way. Probably, the class of structurable algebras [1], which includes Jordan and alternative (with involution) algebras and is deeply related to Lie algebras, is an appropriate candidate, cf.[15].

Some familiarity with basic facts on associative, Lie algebras (e.g., [34, 35, 17]) and varieties of algebraic systems is assumed; although we recall some necessary facts about free algebras, identities, linearizations, etc., for a detailed exposition the reader is referred to [5, 37, 34, 33, 25].

The ground field F will be always of characteristic 0; we will also use the following notations:

∘ vect_{F} (X) = vect(X) is the span of the subset X in a vector space over F; ∘ End(V) = End_{F} (V) is the algebra of linear transformations of a vector space over

F; ∘ F [V] is the algebra of F -valued polynomial functions on a vector space V; ∘ Pol(V, W) ≃ F [V] ⊗_{F} W is the space of polynomial mappings from a vectors space

V to a vector space W; ∘ kV is the direct sum of k copies of a vector space V; V⃗ is the direct sum of a

countable number of V; ∘ If A, B are subspaces of an algebra, then A · B denotes the span of all a · b, where

a ∈ A and b ∈ B; ∘ A^{(−)} is the Lie algebra of an associative algebra A with multiplication [a, b] = ab−ba; ∘ A^{#} is an associative algebra A with the adjoint identity element, A^{#} = F · 1 ˙+A; ∘ ℕ is the set of naturals; ℕ_{0} = ℕ ∪ {0}; ∘ Sym(M) is the symmetric group of a set M.

6 CONTENTS

CHAPTER 1 Kemer’s theorem

This chapter gives a proof of Kemer’s theorem about representability of associative relatively free PI algebras in the case of finite rank. Throughout this chapter ”algebra” means an associative algebra (unless specified); we do not assume that A has the identity element.

1. Algebras of generic elements

We start with some basic definitions , cf. [34, 37]. Let A⟨X⟩ denote the free algebra with the set of free generators X (variables); it means that every mapping π of X to an algebra A extends to an algebra homomorphism from A⟨X⟩ to A. The cardinality of X is called the rank of A⟨X⟩; unless specified, we assume that X is a countable set.

In particular, every algebra endomorphism of A⟨X⟩ is defined by a mapping from X to A⟨X⟩. A subspace of A⟨X⟩ is said to be fully characteristic if it is invariant under all algebra endomorphisms of A⟨X⟩ (i.e., it is invariant under substitution of any polynomials in place of variables).

A polynomial f(x_{1}, …, x_{n}) ∈ A⟨X⟩ is called an identity of an algebra A, if it vanishes under substitution of any elements in A in place of variables. The set of all identities of A forms a fully characteristic ideal of A⟨X⟩, i.e., a T-ideal. Conversely, a T-ideal Γ ⊆ A⟨X⟩ is equal to T (A), where A = A⟨X⟩/Γ. The class of all algebras satisfying a fixed set of identities S is a subvariety 𝔐; let Γ be the T-ideal generated by S, i.e., Γ = (S)_{⊤}, then A⟨X⟩/Γ is the free algebra A_{𝔐}⟨X⟩ of the variety 𝔐; we also call it a relatively free algebra.

Recall that A is said to be a PI-algebra if it satisfies an nontrivial identity; in other words, it means that its ideal of identities T (A) ◁ A⟨X⟩ is not equal to zero [34, 37]. The main objective of this chapter is to prove the following statement due to A.Kemer [20].

Theorem 1.1. Let A be a finitely generated associative PI-algebra. Then there exists a finite dimensional algebra B such that T (A) = T (B).

This fact can rewritten in terms of generic elements.

Let A be an algebra over a field F. Recall that kA is the direct sum of k copies of A; let y_{i} denote the projection of kA onto the i-th summand, i.e., y_{i} is a mapping which takes a vector a ∈ kA to its i-th coordinate, i.e., an element in A. So y_{i} is a function from kA to A; the set of all functions from kA to A (defined at all points) is an algebra with respect to pointwise operations. Let F_{k}(A) denote the subalgebra generated by y_{1}, …, y_{k}, we call it the algebra of generic elements of A (of rank k).

For example, if A is the ground field F (which is infinite), then F_{k}(A) is isomorphic to the algebra of usual polynomials without the absolute term. Next, suppose A is finite dimensional, then y_{i} belongs to the algebra of polynomial functions Pol(kA, A). Let y_{ij} denote the coordinates of y_{i} with respect to a basis e_{1}, …, e_{n} of A, then Pol(kA, A) is isomorphic to the scalar extension of A:

Pol(kA, A) ≃ F [kA] ⊗_{F} A

where F [kA] is the algebra polynomial functions from the vector space kA to A that is isomorphic to the algebra of polynomials F [y_{ij}].

In fact, we need an infinite number of variables, i.e., generic elements; let A⃗ denote the direct sum of a countable number of copies of A, we consider it as the direct limit of kA, so the algebra of generic elements F(A) is generated by y_{1}, …, y_{k}, . . .. It is convenient to assume that there is a one-to-one correspondence between the set of generic elements and the set of variables X.

Lemma 1.1. Let π : A⟨X⟩ → F(A) be the epimorphism defined by the mapping x_{i} → y_{i}, then Ker(π) = T (A).

Proof. Obviously, T (A) ⊆ Ker(π); conversely, if f = f(x_{1}, …, x_{n}) is not an identity of A, then f(a_{1}, …, a_{n}) ≠ 0 for suitable a_{1}, …, a_{n} ∈ A, hence f(y_{1}, …, y_{n}) is a nonzero function, .i.e, π(f) ≠ 0. □

So, F(A) ≃ A⟨X⟩/T (A) and theorem 1.1 can be restated in the following form.

Corollary 1.1. Let A be a finitely generated associative PI algebra. Then there exists a finite dimensional algebra B such that

F(A) ≃ F(B) .

The underlying idea of the proof can be described briefly as follows. It is known that the Jacobson radical of a finitely generated PI algebra is nilpotent [32, 18, 8], hence, by [24], it satisfies all identities of a suitable finite dimensional algebra. Let Γ = T (A), then there exists a f.d. algebra B such that

T (B) ⊆ Γ ;  (1.1)

the idea is to approximate gradually Γ by ideals of identities of finite dimensional algebras. To that end, we estimate the difference Γ\T (B) and, changing B, reduce it to the empty set.

2. Linearization

In this section we give some miscellaneous facts mostly related to linearization of polynomials, cf. [34, 37, 5].

Let V be a vector space; like in the previous section the projector of V⃗ onto the i-th component is called a generic element; let X denote the set of them. Next, let Map(V⃗ , W) be the vector space of all mappings from V⃗ to a vector space W defined at all points of V⃗. For example, X ⊆ Map(V⃗ , V); pick a generic element x_{i} of V and let x_{i}^{−1} : V → V⃗ denote the mapping of V onto the i-th component, so the composition x_{i}^{−1} ∘ x_{i} is the identity mapping of V.

Take f ∈ Map(V⃗ , W) , g ∈ Map(V⃗ , V), then the composition of g, x_{i}^{−1} and f means the substitution of g in place of x_{i} and denoted by f|_{x_{i}=g}. In particular, if g = a is a constant, then the substitution x_{i} → a means that we fix the value of x_{i}.

Take a generic element x ∈ X; a mapping f ∈ Map(V⃗ , W) is said to be homogeneous with respect to x if there is k ∈ ℕ_{0} such that ∀λ ∈ F we have f|_{x=λx} = λ^{k}f. Obviously, for a nonzero f such a k is unique; we call it the degree of f with respect to x and denote by deg_{x}(f). Clearly, if f does not depend on x, then deg_{x}(f) = 0; besides, deg_{x}(fg) = deg_{x}(f) + deg_{x}(g) for all f, g.

Suppose f ≠ 0 is homogeneous with respect to all generic elements; since f belongs to Map(kV, W) for sufficiently big k, the number of x with deg_{x}(f) ≠ 0 is finite, so we may put

deg(f) = ∑_{x} deg_{x}(f) .

Let U be a subspace of Map(V⃗ , W), for each x ∈ X fix k_{x} ∈ ℕ_{0} and consider the set of all homogeneous polynomials of degree k_{x} with respect to x ∈ X. Obviously, it is closed under vector space operations, we call it a homogeneous component of V (of degree k_{x} with respect to x ∈ X).

A subspace U is said to be homogeneous ,if it is spanned by homogeneous mappings; using Vandermonde determinant argument, e.g., [37, p. ], one can show that U is the direct sum of its homogeneous components. Moreover, in the same way one can check the following statement.

Lemma 1.2. Let U be a subspace of a homogeneous space U′ ⊆ Map(V⃗ , W) such that for all f ∈ U and λ ∈ F we have f_{x=λ_{x}} ∈ U. Then U is homogeneous too.

Let f ≠ 0 be a homogeneous mapping in a homogeneous space U ⊆ Map(V⃗ , W), denote by varb(f) the set of all x such that deg_{x}(f) ≠ 0; take an arbitrary nonzero f ∈ U and let f = ∑_{i=1}^{k} f_{i} be the decomposition into the sum of homogeneous components, then varb(f) is defined as ⋃_{i=1}^{k} varb(f_{i}).

Next, let U be a S-subspace of Map(V⃗ , W), i.e., a homogeneous subspace which is invariant under substitution of linear combination of generic elements. We will define the opeartor of linearization L_{x}(g) for x ∈ X and g ∈ Map(V⃗ , V), cf. [37, p.9].

Pick f and take y ∈ X such that deg_{y}(f) = 0 and consider the decomposition of f|_{x=x+y} into the sum of homogeneous components with respect to y,

f|_{x=x+y} = ∑_{i=0}^{∞} f_{i} ,  (1.2)

where f_{i} is homogeneous in y of degree i. Set f · L_{x}(g) = f_{1}|_{y=g}; since U is an S-subspace, this element belongs to U thanks to the lemma above; note that the definition does not depend on y. If deg_{x}(f) = k ≠ 0, then the degree of f · L_{x}(y), where y ∈ X and deg_{y}(f) = 0, is equal to deg_{x}(f) = k − 1. Moreover,

f · L_{x}(y)|_{y=x} = k · f .  (1.3)

Indeed, for all λ ∈ F

(1 + λ)^{k}f = f|_{x=(1+λ)x} = (f|_{x=x+y})|_{y=λx} = ∑_{i=0}^{∞} λ^{i}f_{i} ,

matching the coefficients at powers of λ in the both sides, we get C_{k}^{i}f = f_{i}.

Now, suppose that U′ is generated as an S-subspace by multilinear mappings (i.e., mappings which are linear with respect to every x ∈ X), then an S-subspace U is homogeneous and also generated by multilinear mappings. Indeed, U is homogeneous by the lemma above and, therefore, it is invariant under operators of partial linearization. Moreover, if deg_{x}(f) = 1, then f is linear with respect to x; taking into account (1.3), we can replace every homogeneous generator of U by its linearizations.

For example, let V, W be an algebra A and let U′ be the algebra of generic elements F(A); clearly, it is a homogeneous S-subspace generated by multilinear mappings, hence, every S-subspace (e.g., a T-ideal ) is generated by multilinear elements; thanks to lemma 1.1 the same fact is valid in relatively free algebras. Another example is the vector space of polynomial mappings Pol(V⃗ , W) ⊆ Map(V⃗ , W); clearly, it is a S-subspace generated by multilinear mappings.

The symmetric group Sym(X) acts naturally on Map(V⃗ , W) by permutation of variables. by definition, every S-subspace is invariant under Sym(X). Using the technique of Young diagrams, e.g. [9], one can prove the following fact, cf. [33, 5, 13].

Proposition 1.1. Let U be a homogeneous S-subspace of Map(V⃗ , W) generated by multilinear mappings. Fix n ∈ ℕ and suppose that every f ∈ U, which is linear and skewsymmetric with respect to n variables, is equal to 0. Then U is generated as an S-subspace by U ∩ Map(nV, W).

In particular, U must be equal to 0, if all mappings in U depending only on x_{1}, …, x_{n} are equal identically to 0.

For example, let A be an algebra satisfying the Capelli identity of rank n + 1

∑_{σ∈§_{n+1}} sgn(σ)x_{σ(1)}y_{1} …, y_{n}x_{σ(n+1)} ,  (1.4)

where S_{n+1} = Sym({1, …, n+1}), cf.[5, 34]. Then T (A) = T (A′), where A′ is an algebra generated by n elements.

Indeed, let Cap_{n} denote T-ideal generated by (1.4); it contains every polynomial in A⟨X⟩ which is linear and skew-symmetric with respect to n variables. By hypothesis, Cap_{n+1} ⊆ T (A), hence, from the proposition above it follows that modulo Cap_{n+1} the S-subspace T (A) is generated by T (A) ∩ A⟨X_{n}⟩, where X_{n} = {x_{1}, …, x_{n}}. Put A′ = A⟨X_{n}⟩/T (A) ∩ A⟨X_{n}⟩, we claim that T (A′) = T (A). Indeed, A′ is embedded into A⟨X⟩/T (A), hence T (A) ⊆ T (A′). In particluar, Cap_{n+1} ⊆ T (A′), hence T (A′) is generated by T (A′) ∩ A⟨X_{n}⟩ by the proposition above. Take f ∈ T (A′) ∩ A⟨X_{n}⟩, it vanishes under the substitution of x_{i}/T (A) ∩ A⟨X_{n}⟩ in place of x_{i}, which means that f ∈ T (A) ∩ A⟨X_{n}⟩; hence, T (A′) = T (A).

3. Specializations of noncommutative polynomials

We begin to prove theorem 1.1.

First of all, we can assume that the ground field F is algebraically closed. Indeed, let F̄ be the algebraic closure of F and let Ā = F̄ ⊗_{F} A, the scalar extension of A by F̄. Besides, F̄ ⊗_{F} A⟨X⟩ is the free algebra over F̄; since F is infinite, every T-ideal is homogeneous by lemma 1.2, therefore the algebras A and Ā have the same identities over F, cf. [37, p.8] Hence, the ideal of identities of Ā over F̄ is equal to F̄ ⊗_{F} T (A).

By hypothesis, T (Ā) = T (C) where C is a finite dimensional algebra over F̄; take a basis c_{1}, …, c_{n} of C and let γ_{ij}^{k} be the structure constants of C,

c_{i}c_{j} = ∑_{k=1}^{n} γ_{ij}^{k}c_{k} .

Let K be the field extension of F by γ_{ij}^{k}, since F̄ is algebraic over F, it is a finite extension. Hence, the span B of c_{1}, …, c_{n} over K is a subalgebra of C over F of finite dimension. Cleraly, in A⟨X⟩ we have T (B) = T (C) = T (Ā) = T (A), as required.

Let A be a finite dimensional algebra over F. We fix a decomposition of A into the sum of the radical R = R(A) and a semisimple subalgebra P ≃ P(A) = A/R(A) [35, p.192]. The algebra P(A) is the direct sum of simple algebras; let pr(P) denote the set of projectors onto simple summands; if α ∈ pr(P), then P_{α} means the corresponding simple subalgebra. Put

Spl(A) = R ∪ ⋃_{α∈pr(P)} P_{α} .  (1.5)

Pick a multilinear polynomial f(x) = f(x_{1}, …, x_{n}) ∈ A⟨X⟩ of degree n, where x means the n-tuple of variables (x_{1}, …, x_{n}) (here the subindices are not related to any fixed enumeration of elements of X).

Consider a substitution x → x̄ of elements A in place x ∈ varb(f), where

x̄ ∈ Spl(A) ;  (1.6)

the n-tuple x̄ is said to be a specialization of x and f̄ = f(x̄) = f(x̄_{1}, …, x̄_{n}) is a specialization of f in A. Put

V(f, A) = {x̄| f(x̄) ≠ 0}  (1.7)

where x̄ runs over all specializations of x in A.

Take a T-ideal Γ ⊈ T (A); to estimate the difference Γ\T (A) we consider the set

V(Γ, A) = ⋃_{f} V(f, A) ,

where f runs over the set of all multilinear polynomials in Γ. Since Γ is generated by multilinear polynomials, V(Γ, A) is empty iff Γ ⊆ T (A).

Type t(Γ, A).

For x̄ = (x̄_{1}, …, x̄_{n}) ∈ V(f, A) put

π(x̄) = {α ∈ π(A)| ∃i x̄_{i} ∈ P_{α}}  (1.8)

and

a(x̄) = ∑_{α∈π(x̄)} dim(P_{α}) .  (1.9)

Next, let b(x̄) denote the number of x_{i} such that x̄_{i} ∈ R,

b(x̄) = |{x_{i}| x̄_{i} ∈ R}| ;  (1.10)

finally, put t(x̄) to be the pair (a(x̄), b(x̄)) in the well-ordered set ℕ_{0} × ℕ_{0} with the lexicographical order: (n, m) ≥ (n′, m′) if n > n′ or n = n′ and m ≥ m′.

For a multilinear f ∈ Γ\T (A) put

t(f, A) = min{t(x̄)| x̄ ∈ V(f, A)} ;  (1.11)

if Γ ⊈ T (A) we can define the type t(Γ, A) as the maximum of t(f, A), where f runs over the subspace of all multilinear polynomials in Γ\T (A). By definition, t(Γ, A) is a pair (a, b) ∈ ℕ_{0} × ℕ_{0}; put a(Γ, A) = a and b(Γ, A) = b,

It turns out that the type t(Γ, A) can be described in terms of identities.

T-ideal S_{a,b,r}(Γ).

Take a, b ∈ ℕ_{0}, r ∈ ℕ and a multilinear polynomial f ∈ Γ, which contains disjoint subsets Λ_{1}, …, Λ_{b} and Θ_{1}, …, Θ_{r} in varb(f) such that

|Λ_{i}| = a + 1 ,  (1.12)
|Θ_{j}| = a  (1.13)

for all i ∈ {1, …, b} and j ∈ {1, …, r}.

Denote by h the polynomial obtained from f by alternating variables in each Λ_{i} and Ω_{j}. More precisely, if M′ is a subset of M, then Sym(M′) is embedded naturally in Sym(M) (all permutations of M′ extend trivially to M). Let G be the product of subgroups in Sym(X):

G = ∏_{j=1}^{r} Sym(Θ_{j}) × ∏_{i=1}^{b} Sym(Λ_{i}) ;  (1.14)

it is the direct product, because the corresponding subsets are disjoint. It acts on A⟨X⟩ by permutation of variables and h = S_{G} · f, where

S_{G} = ∑_{σ∈G} sgn(σ)σ.  (1.15)

So, S_{a,b,r}(Γ) denotes the T-ideal generated by all h defined in this way.

Proposition 1.2. Let A be a f.d. algebra and let Γ be a T -ideal, Γ ⊈ T (A). Let S(Γ, A) denote the set of all (a, b) ∈ ℕ_{0} × ℕ_{0} such that

S_{a,b,r}(Γ) ⊈ T (A)  (1.16)

for all r ∈ ℕ. Then

S(Γ, A) = {(a, b) ∈ ℕ_{0} × ℕ_{0}| (a, b) ≤ t(Γ, A)} .  (1.17)

Proof. In particular, the proposition says that t(Γ, A) is the greatest element of S(Γ, A); we claim that this statement is equivalent to (1.17). Indeed, suppose (a, b) = t(Γ, A) ∈ S(Γ, A) and take any (a′, b′) < (a, b); by definition, for all a′ < a and b′ ∈ ℕ_{0} we have

S_{a,b,r+b′}(Γ) ⊆ S_{a′,b′,r}(Γ)

hence, in this case (a′, b′) ∈ S(Γ, A). Next, if b′ < b, then

S_{a,b,r}(Γ) ⊆ S_{a,b′,r}(Γ)

hence, the same holds if a = a′.

Now, suppose that (a, b) ∈ S(Γ, A), let us show that (a, b) ≤ t(Γ, A). Let r ∈ ℕ such that R^{r} = 0, by hypothesis there is a multilinear h ∈ S_{a,b,r}(Γ) which is not an identity of A; we may assume that it is equal to S_{a,b,r}(f) for suitable f ∈ Γ. So, the set V(h, A) is not empty, take x̄ such that t(h, A) = (a(x̄), b(x̄)).

First, suppose a > a(x̄); since h is skew-symmetric with respect to variables in Θ_{j} and |Θ_{j}| = a, there is x ∈ Θ_{j} such that x̄ ∈ R. Hence, f(x̄) ∈ R^{r} = 0 which contradicts to the hypothesis. So, a ≤ a(x̄); if the inequality is strict, then (a, b) < t(h, A) ≤ t(Γ, A), as required. Thus, we assume a = a(x̄); hence, for all i ∈ {1, …, b} there is x ∈ Λ_{i} such that x̄ ∈ R, hence b(x̄) ≤ b and (a, b) ≤ t(h, A) ≤ t(Γ, A).

Now we need only to prove that t(Γ, A) ∈ S(Γ, A); let a = a(Γ, A), b = b(Γ, A), we need only to show that (1.16) holds for all r.

Lemma 1.3. Let P be a simple algebra and let f = f(x_{1}, …, x_{n}) ∉ T (P). For all a ∈ P there exists x̄_{i}, b_{j}, c_{j} ∈ P such that

a = ∑_{j=1}^{m} b_{j}f(x̄_{1}, …, x̄_{n})c_{j} .  (1.18)

Proof. Take x̄_{i} ∈ P such that f̄ = f(x̄_{1}, …, x̄_{n}) ≠ 0, then the ideal generated by f̄ is equal to P. On the other hand, it is equal to P f̄P. □

It is well-known [34, p.23] that Cap_{t^{2}} is not contained in the ideal of identities of the algebra of t × t matrices M_{t}(F), hence any a ∈ M_{t}(F) is represented in the form (1.18) where f is skew-symmetric with respect to t^{2} variables. Moreover, for all r ∈ ℕ the T-ideal Cap_{t^{2}}^{r} ⊈ T (M_{t}(F)), hence, we can find f with disjoint family of subsets λ_{1}, …, λ_{r} ⊆ varb(f) such that f is skew-symmetric with respect to each λ_{j}.

By the previous argument, for each α ∈ pr(P) there exists a polynomial h_{α} such that (*) a specialization of h_{α} in P_{α} is equal to the identity element e_{α} of P_{α}; (**) there are disjoint subsets of variables λ_{1}^{α}, …, λ_{r+b}^{α} such that h_{α} is multilinear and

skew-symmetric with respect to each λ_{α}^{j} and |λ_{j}^{α}| = dim(P_{α}) for all j. Take a multilinear f = f(x) ∈ Γ such that t(Γ, A) = t(f, A) and let x̄ ∈ V(f, A) such

that t(x̄) = t(f, A); recall it is the least one among t(x̃), x̃ ∈ V(f, Γ). Hence, for any specialization x → x̃ of variables of f with t(x̃) < t(x̄) we have f(x̃) = 0. Denote by Z(f, x̄) the set {z_{1}, …, z_{b}} of all variables in varb(f) such that z̄_{i} ∈ R.

First, conisder the case when π(x̄) contains a single projector α and f̄· e_{α} (or e_{α} · f̄) is not zero. Since F is algebraically closed, P_{α} is isomorphic to a matrix algebra M_{t}(F), where a = t^{2}. By the lemma above, for all r ∈ ℕ there is a polynomial h = h_{α} satisfying (*), (**); put f_{0} = f · h. We can assume varb(h) ∩ varb(f), so this substitution extends x̄; denote it by x → x̄, thus f̄_{0} = f̄· h̄ = f̄ ≠ 0.

For all i ∈ {1, …, b} and j ∈ {1, …, r} put Λ_{i} = {z_{i}} ∪ λ_{i}^{α}, Θ_{j} = λ_{j+b}^{α} and let

g = S_{G}(f_{0}) ,

where G is the group (1.14). Clearly, g ∈ S_{a,b,r}(Γ), so it suffices to show that g does not vanish under x → x̄.

Consider \overline{σ · f_{0}}, where σ ∈ G; suppose σ(z_{j}) ≠ z_{j} for some j, then it is equal to f̃· h̃, where a(x̃) = a(x̄) and |Z(f, x̃)| < b. Hence, t(x̃) < t(x̄) and therefore \overline{σ · f_{0}} = 0, hence, ḡ = |G_{1}| · f̄h̄ ≠ 0, where

G_{1} = ∏_{i=1}^{r+b} Sym(λ_{i}^{α}) ≤ G .

Now consider the general case; recall that e_{α} denotes the identity element of P_{α}; put

e = ∑_{α∈π(x̄)} e_{α}

and e_{0} = 1 − e ∈ A^{#} = F ˙+A. It defines the Peirce decomposition of R with respect to the orthogonal system of idempotents e_{0}, e_{α}, e.g. [35, p.35],

R = ∑_{α,β∈π(x̄)^{#}} R_{α,β} ,

where R_{α,β} = e_{α}Re_{β} and π(x̄)^{#} = π(x̄) ∪ {0}. Obviously, we can assume that all x̄ from R lie in the Pierce components of R; besides, by the previous case, we can assume that |π(x̄)| > 1 or e_{0}f̄e_{0} = (1 − e)f(1 − e) = f̄ ≠ 0.

By definition, for every α ∈ π(x̄) there exists y ∈ varb(f) such that ȳ ∈ P_{α}; hence there exists z ∈ Z(f, x̄) such that

z̄ ∈ R_{α,β} ∪ R_{β,α} ,  (1.19)

where β ≠ α; fix such z = z_{α} ∈ Z(f, x̄), in this case we say that z is marked for α. Denote by m(z) the set of α ∈ π(x̄) such that z is marked for α; obviously, 0 ≤ |m(z)| ≤ 2.

By lemma 1.3, for all α there is a polynomial h_{α} satisfying (*), (**); put

h′_{α} = x_{α}h_{α}y_{α}

with x̄_{α} = e_{α} and ȳ_{β} = e_{β} and to every z ∈ Z(h, x̄) define a polynomial ϕ_{z}:

(1) if m(z) = ∅, then ϕ_{z} = z;

(2) let m(z) = {α}; then z̄_{α} ∈ R_{β,α} (or z̄_{α} ∈ R_{α,β}) for β ≠ α, put ϕ_{z} = z′zh′_{α} (ϕ_{α} = h′_{α}zz′, respectively) with z̄′ = e_{β};

(3) if m(z) = {α, β} and z̄ ∈ R_{α,β}, then ϕ_{z} = h′_{α} · z · h′_{β} . Next, put

f_{0} = f|_{∀z z=ϕ_{z}} ,

where z runs over Z(f, x̄); we may assume that the variables envolved in every ϕ_{z} and varb(f)\Z(f, x̄) form disjoint sets, so the substitution x → x̄ extends to that for all ϕ_{z} and gives ψ̄_{z} = z̄, therefore f̄_{0} = f̄ ≠ 0. Join variables letting

Λ_{i} = {z_{i}} ∪ ⋃_{α∈π(x̄)} λ_{i}^{α} ,

Θ_{j} = ⋃_{α∈π(x̄)} λ_{b+j}^{α} ,

for all i ∈ {1, …, b} and j ∈ {1, …, r}; by definition |Λ_{i}| = a(x̄) + 1 = a + 1 and |Θ_{j}| = a + 1. Put g = S_{G} · f_{0}, where G is the group (1.14), then g ∈ S_{a,b,r}(Γ), so it suffices to check that ḡ ≠ 0.

Take σ ∈ G and suppose that \overline{σ · f_{0}} ≠ 0, we claim that σ(z) = z for all z = z_{i} ∈ Z(f, x̄) and σ(λ_{j}^{α}) = λ_{j}^{α} for all j, α. First, consider z with m_{z} ≠ ∅ and suppose σ(z) ≠ z, then σ(z) ∈ λ_{i}^{α} and \overline{σ(z)} ∈ P_{α}. Recall that the variable z in ϕ_{z} is surrounded by two variables being fixed under σ and taking value in different simple components of P under x → x̄. Hence, \overline{σ · ϕ_{z}} = 0 and therefore \overline{σ · f_{0}} = 0. Next, take z with m(z) = ∅ and suppose σ(z) ≠ z, then σ^{−1}(z) = x ∈ λ_{i}^{α} for some i, α. There exists z′ ∈ Z(f, x̄) such that α ∈ m(z′); by the previous, σ(z′) = z′, hence \overline{σ · f_{0}} = f(x̃), where z̃′ ∈ R^{2}. Hence, b(x̃) < b(x̄); besides, a(x̃) = a(x̄), hence t(f, x̃) < t(f, x̄) and f(x̃) = 0.

So, σ(z) = z for all z ∈ Z(f, x̄); now, suppose that σ(x) ∈ λ_{i}^{α} for some x ∈ λ_{i}^{β}, β ≠ α. Observe that

σ · h′_{α} = x_{α}(σ · h_{α})y_{α}

and all variables involved in σ · h_{α} take value in P, hence \overline{σ · h′_{α}} = 0. So σ(λ_{i}^{α}) = λ_{i}^{α} for all α, i; hence ḡ = q · f̄_{0} for some q ∈ ℕ. The proposition is proven. □

4. Decomposition of ideals of identities

We will need a general construction of finite dimensional algebra based on free products of algebras, e.g., [5]. Recall that an algebra A ∗ B is called a free product of algebras A, B if

(i) A, B are subalgebras of A ∗ B;

(ii) A ∗ B is generated by A and B; (iii) if ϕ : A → D, ψ : B → D are homomorphisms of algebras, then there is a

homomorphism π : A ∗ B → D such that π|_{A} = ϕ and π|_{B} = ψ. Now, let P be a finite dimensional algebra and let Y be a finite set of variables, |Y | = d.

Denote by P ⟨Y ⟩ the free product of P and the free algebra A⟨Y ⟩. By (iii), any mapping π : Y → P ⟨Y ⟩ can be extended to an endomorphism π̃ of P ⟨Y ⟩ such that π̃|_{P} = id_{P}. In terms of variables it means that we may replace variables from Y by elements in P ⟨Y ⟩. Recall that an element f ∈ P is homogeneous with respect to y ∈ Y of degree m, if for every λ ∈ F we have f|_{y=λy} = λ^{m}f. Denote by P ⟨Y ⟩^{(i)} the subspace of homogeneous elements in P ⟨Y ⟩ of total degree i; in particular, P ⟨Y ⟩^{(0)} = P.

Clearly, algebra P ⟨Y ⟩ has the following properties:

(i) P ⟨Y ⟩ is the direct sum of its homogeneous components,

P ⟨Y ⟩ = ⨁_{i=0}^{∞} P ⟨Y ⟩^{(i)} ;

(ii) for all i the subspace P ⟨Y ⟩^{(i)} is finite dimensional.

Let Γ be a T -ideal of A⟨X⟩; denote by I the verbal ideal of Γ in P ⟨Y ⟩,

I = Γ[P ⟨Y ⟩] = {f(a_{1}, …, a_{n})| f ∈ Γ; a_{i} ∈ P ⟨Y ⟩} ;

also, let J be the ideal generated by Y; take b ∈ ℕ_{0} and put

K(P, Y, Γ, b) = P ⟨Y ⟩/V  (1.20)

where V = I + J^{b+1}.

Lemma 1.4. The algebra K(P, Y, Γ, b) is finite dimensional and satisfies all identities in Γ. Besides, if Γ ⊆ T (P), then P is isomorphic to P + V/V ⊆ K(P, Y, Γ, b).

Proof. The first conjecture follows from (i) above and the definition. Next, since Γ is homogeneous, I and J^{b+1} are homogeneous with respect to every variable from Y, hence P ∩ V ⊆ P ∩ I; moreover, I ∩ P = Γ[P] = 0 because Γ ⊆ T (P), hence P ≃ P + V/V ⊆ K(P, Y, Γ, b). □

Let A be a finite dimensional algebra; denote by a(A) the dimension of P(A), also let b(A) ∈ ℕ_{0} such that b(A)+1 is the nilpotency index of R(A); finally t(A) = (a(A), b(A)).

Proposition 1.3. Let A be a f.d. algebra. For all a, b ∈ ℕ_{0} there exists a finite family of algebras B^{ξ} (ξ ∈ Ξ) such that

(a) T (A) ⊆ T (B^{ξ}) for all ξ ∈ Ξ;

(b) t(B^{ξ}) ≤ (a, b) for all ξ ∈ Ξ;

(c) for all multilinear f ∈ A⟨X⟩, if f ∉ T (A) and t(f, A) ≤ (a, b), then

f ∉ ⋂_{ξ∈Ξ} T (B^{ξ}) .

Proof. Let Ξ denote the set of all subsets ξ ⊆ pr(P), where P = P(A), such that ∑_{α∈ξ} dim(P_{α}) ≤ a. Put P^{ξ} = ∑_{α∈ξ} P_{α} and B^{ξ} = K(P^{ξ}, Y, T (A), c), where |Y | = dim(R) and

c = \{ \begin{matrix}b & , dim(P^{ξ}) = a \\ b(A) & , otherwise. \end{matrix}

By lemma 1.4 the algebra B^{ξ} is finite dimensional and T (A) ⊆ T (B^{ξ}); also, T (A) ⊆ T (P^{ξ}), hence P^{ξ} is a subalgebra of B^{ξ}. The radical R(B^{ξ}) is generated by Y/V (in notations of lemma 1.4), hence P^{ξ} is a semisimple part of B^{ξ}. Thus, t(B^{ξ}) ≤ (a, b).

Let us prove (c); take x̄ ∈ V(f, A) such that t(x̄) = t(f, A); then |π(x̄)| = a(x̄) ≤ a and ξ = π(x̄) belongs to J. Define a mapping from Y onto a basis of R; it extends to a homomorphism ψ from K^{ξ} = P^{ξ}⟨Y ⟩ to A, where ψ restricts to the identity mapping on P^{ξ}. By definition, for all x ∈ varb(f) the specialization x̄ lies in the image of ψ. Denote by x̃ ∈ K^{ξ} a pre-image of x̄, where x̃ ∈ vect(Y) provided x̄ ∈ R and x̃ = x̄ if x̄ ∈ P^{ξ}. Then the specialization f̃ = f(x̃_{1}, …, x̃_{m}) is not equal to 0 and, since b(x̄) ≤ b, its degree with respect to Y is ≤ c. Besides, the verbal ideal Γ[K^{ξ}] lies in the kernel of ψ, hence f̃ ∉ Γ[K^{ξ}] and therefore f̃ ∉ Γ[K^{ξ}] + J^{c+1} and f ∉ T (B^{ξ}). □

An algebra A is called representable if there exists an extension K of the ground field F and a finite dimensional algebra A′ over K such that A is embedded in A′ as an algebra over F. Let B ⊆ A′ denote the span of A over K, then T (B) = T (A). Indeed, since A ⊆ B, we have T (B) ⊆ T (A); on the other hand, T (A) is generated by multilinear polynomials, hence T (A) ⊆ T (B).

Lemma 1.5. Let B an algebra over F that is finite dimensional over an extension K of F. Then there exists a finite dimensional algebra C over F such that T (C) = T (B).

Proof. Recall that B is decomposed into the sum of the radical R(B) and a semisimple subalgebra P(B); besides, we may assume that H is algebraically closed, hence P (B) is the direct sum of matrix algebras, and therefore P(B) ≃ K ⊗_{F} P, where P is a direct sum of matrix algebras over F.

Next, put

C = K(P, Y, T (B), b(B))

where |Y | = dim_{K}(R(B)). By lemma 1.4, C is a finite dimensional algebra over F and T (B) ⊆ T (C). Now suppose that the reciprocal inclusion is untrue, then there is a multilinear polynomial f(x_{1}, …, x_{n}) ∈ T (C) which is not an identity of B. Let f̄ ≠ 0 be a specialization of f in B, moreover , we may assume that x̄_{i} are elements of a basis of B. Hence, we can define a homomorphism π from P ⟨Y ⟩ to B which takes elements from Y to the basis elements of the radical R(B) and restricts to the identity mapping of P ⊆ P(B). Hence, there is f̃ ∈ π^{−1}(f̄) being homogeneous with respect to Y and degree ≤ b, hence, f̄ ∉ V and f̂ = f̃/V is a nonzero specialization of f in C which is impossible. □

So, to prove theorem 1.1 it suffices to show that A⟨X⟩/T (A) is representable.

5. Specht’s conjecture

A positive solution of Specht’s problem was given by A.Kemer in [19]; it says that any associative algebra over a field of characteristic 0 has a finite basis of identities. In fact, the proof consists of two parts; the first one shows that the variety of any PI algebra is generated by the Grassmann envelope of a finitely generated PI superalgebra, and the second part states d.c.c. conditions for subvarieties. In this section we gives the argument of the second part adopted to the case of algebras which gives Specht’s conjecture for finitely generated algebras.

Corollary 1.2. The ideal of identities of a finitely generated algebra A over F is a finitely generated T-ideal.

Suppose it is untrue; we assume that T (A) ≠ 0, hence A is a finitely generated PI algebra. By [18], Cap_{n} ⊆ T (A) for some n; besides, by proposition 1.1 Cap_{n} is the ideal of identities of an algebra generated by n − 1 elements; besides, by [32, 18, 8], the Jacobson radical of of a finitely generated PI algebra is nilpotent, hence, thanks to [24], Cap_{n} contains the ideal of identities of a finite dimensional algebra B. Since Cap_{n} is a finitely generated T-ideal, it suffices to show that T (A) is finitely generated modulo T (B).

Proposition 1.4. Let A be a finite dimensional algebra, then modulo T (A) any T-ideal is generated by a finite number of polynomials.

Proof. Suppose it does not hold; since every T-ideal is generated by multilinear polynomials, there exists a sequence of multilinear polynomials f_{n}, n = 1, …, ∞, such that deg(f_{n}) < deg(f_{n+1}) for all n and f_{n+1} ∉ T (A) + (f_{1}, …, f_{n})_{⊤}.

Put

(f_{n})_{⊤}^{+} = (f_{n}x, xf_{n}, f_{n}|_{x_{i}=x_{i}x}, f_{n}|_{x_{i}=xx_{i}}| ∀i, f_{n} = f_{n}(x_{1}, …, x_{i}, …, x_{n}), ∀x ∈ X)_{⊤} .

In other words, it is the T-ideal of all ”proper” consequences of the identity f_{n}; let Γ = T (A) + ∑_{n=1}^{∞}(f_{n})_{⊤}^{+}.

From homogeneity of T -ideals it follows that f_{n} ∉ Γ for all n, but (f_{n})_{⊤}^{+} ⊆ Γ. We call such a sequence of polynomials {f_{n}| n ∈ ℕ} appropriate. A T-ideal which has an appropriate sequence is called critical.

Proposition 1.5. Let Γ be a critical T -ideal, then Γ ≠ T (C), where C is a finitedimensional algebra.

Proof. On the contrary, suppose that Γ = T (C), where C is a finite dimensional algebra; recall that C is the sum of the radical R(C) and a semisimple subalgebra P(C). Take an element f_{n} of the appropriate sequence such that deg(f_{n}) ≥ b(C); since f_{n} = f_{n}(x_{1}, …, x_{k}) ∉ Γ = T (C), there are a_{1}, …, a_{k} ∈ C such that f̄_{n} = f_{n}(a_{1}, …, a_{k}) ≠ 0.

Recall that f_{n} is multilinear, therefore, we may assume that a_{i} ∈ R(C) ∪ P(C) for all i. Suppose that a_{j} ∈ P(C) for some j; by the definition, f′_{n}(x_{1}, …, x_{k+1}) = f_{n}|_{x_{j}=x_{j}x_{k+1}} ∈ Γ = T (C), hence, f̄_{n} must be zero. On the other hand, let e be the identity element of P(C), then f′_{n}(a_{1}, …, a_{k}, e) = f̄_{n}. So, a_{i} ∈ R(C) for all i, but in this case f̄_{n} also vanishes, because deg(f_{n}) ≥ c. □

From theorem 1.1 it follows that critical T-ideals do not exist, hence, it proves Specht’s conjecture. However, we give a shorter proof of this fact.

”Counter-example”.

Call a pair of T-ideals

T (A) ⊂ Γ

a ”counter-example”, if A is a finite-dimensional algebra and Γ is a critical T -ideal.

Now suppose that a ”counter-example” exists, then there exists a ”counter-example” T (A) ⊂ Γ of the smallest type t(Γ, A).

Lemma 1.6. Let Γ, Γ′ be T-ideals, where Γ is critical. Then Γ+Γ′ or Γ∩Γ′ is critical.

Proof. Let {f_{i}| i = 1, 2, … } be the appropriate sequence for Γ. Suppose that every subsequence is not appropriate for Γ + Γ′, then there exists n such that for all m ≥ n we have f_{m} = h_{m} + g_{m}, where h_{m} ∈ Γ, g_{m} ∈ Γ′ are multilineal polynomials of degree deg(f_{m}). Since f_{m} ∉ Γ and h_{m} ∈ Γ, we get g_{m} ∉ Γ. Besides, (g_{m})_{⊤}^{+} ⊆ (f_{m})_{⊤}^{+} + (h_{m})_{⊤}^{+} ⊆ Γ and therefore

(g_{m})_{⊤}^{+} ⊆ Γ ∩ Γ′ .

Hence, {g_{n}| n ≥ m} is an appropriate sequence for Γ ∩ Γ′. □

Lemma 1.7. Let Γ ⊆ T (C) be a ”counter-example”. Suppose there is a family of finite dimensional algebras C_{i}, i = {1, …, k} such that

T (C) = Γ ∩ ⋂_{i=1}^{k} T (C_{i}) .

Then for some i there exists a ”counter-example” T (C_{i}) ⊂ Γ′, where Γ′ ⊆ Γ + T (C_{i}).

Proof. Put Υ_{0} = Γ and

Υ_{m} = Γ ∩ ⋂_{i=1}^{m} T (C_{i}) ;

take the maximal m ∈ {0, …, k} such that Υ_{m} is critical, then m < k and by the previous lemma,

Γ′ = Υ + T (C_{m+1})

is critical. □

From proposition 1.3 we get immediately

Corollary 1.3. There exists a counter example T (B) ⊂ Γ′, where t(B) ≤ t(Γ, A).

Since t(Γ, A) is minimal and t(Γ′, B) ≤ t(B), we have t(Γ′, B) = t(B). In the next section we will exhibit a ”counter example” of lesser type.

6. Finite-dimensional embedding

Let A be a finite dimensional algebra and let tr(x) be a linear mapping to the centroid C(A) of A,

C(A) = {ϕ ∈ End(A)| ∀a, b ∈ A ϕ(ab) = ϕ(a)b = aϕ(b)} .

Denote by Ψ the subalgebra of Pol(A⃗, C(A)) generated by 1 and tr(u), where u runs over F(A) and let Ψ·F(A) denote the Ψ-module generated by Ψ in Pol(A⃗, A). Clearly, Ψ·F(A) is a fully characteristic ideal, i.e., it is invariant under substitution of elements from F(A) in place of variables (generic elements).

We say F(A) is integral over Ψ if for all f ∈ F(A) there are ϕ_{1}, …, ϕ_{n} ∈ Ψ such that

f^{n+1} = ∑_{i=1}^{n} ϕ_{i}f^{i} .  (1.21)

Lemma 1.8. Suppose that F(A) is integral over Ψ. Let I be a fully characteristic ideal of Ψ · F(A), then the full pre-image of I ∩ F(A) with respect to the canonical homomorphism fron A⟨X⟩ onto F(A), see proposition 1.1, is equal to the ideal of identities of a finite dimensional algebra C,

I ∩ F(A) = T (C)/T (A) .

Proof. Let Υ be the full pre-image of I ∩ F(A), i.e., Υ/T (A) = I ∩ F(A); since I is fully characteristic, Υ is a T-ideal. Let k = dim(A), then Cap_{k+1} ⊆ T (A) ⊆ Υ; by lemma 1.1 Υ is the ideal of identities of A⟨X_{k}⟩/Υ_{k}, where Υ_{k} = Υ ∩ A⟨X_{k}⟩. Thanks to lemma 1.5 it suffices to show that A⟨X_{k}⟩/Υ_{k} is representable.

Recall that F_{k}(A) is the subalgebra of F(A) generated by x_{1}, …, x_{k}; in fact, it is the span of all elements in F(A) of zero degree with respect to all x ∉ {x_{1}, …, x_{k}}. Let Ψ_{k} denote the span of such elements in in Φ; clearly, it is generated by tr(u), where u ∈ F_{k}(A). Put B_{k} = Ψ_{k} · F_{k}(A); it is the corresponding homogeneous component of Ψ · F(A); besides,

Υ_{k} + T (A)/T (A) = I ∩ F_{k}(A) = I_{k} ∩ F_{k}(A)

where I_{k} = I ∩ B_{k}. Hence, A⟨X_{k}⟩/Υ_{k} is embedded into B_{k}/I_{k} and we need only to show that the latter algebra is representable.

By (1.21) every element in F_{k}(A) is integral over Ψ; thanks to homogeniety, we can assume that all ϕ_{j} in (1.21) are in Ψ_{k}, so F_{k}(A) is integral over Ψ_{k}, Besides, B_{k} inherits all identities of A, in particular, it is PI; finally, it is generated by x_{1}, …, x_{k} over Φ_{k}, hence, by Shirshov’s theorem on hieght [34, 37], B_{k} is finitely generated Ψ_{k}-module, hence so is B_{k}/I_{k}. Let u_{1}, …, u_{m} be the generators, then Ψ_{k} as an algebra over F is generated by 1 and tr(u_{1}), …, tr(u_{m}). From [7, 3] it follows that B_{k}/I_{k} is representable. □

Now we continue the proof of theorem 1.1. Let t(Γ, A) = (a, b), by proposition 1.3 there is a finte family of finite dimensional algebras B^{ξ}, ξ ∈ Ξ, such that t(B) ≤ t(Γ, A).

Proposition 1.6. There exists r_{0} ∈ ℕ such that for all r ≥ r_{0} there exists a f.d. algebra B such that

T (B) = S_{a,b,r}(Γ) + ⋂_{xi∈Ξ} T (B^{ξ}) .  (1.22)

Proof. Take ξ ∈ Ξ such that t(B^{ξ}) < t(Γ, A), then t(Γ, B^{ξ}) < (a, b) by proposition 1.2. Hence, there exists r ∈ ℕ such that

S_{a,b,r}(Γ) ⊂ T (B^{ξ}) ;  (1.23)

since the set Ξ is finite, we can assume that it holds for all ξ ∈ Ξ with t(B^{ξ}) < (a, b).

Consider the direct sum of algebras C = ⨁_{ξ∈Ξ} B^{ξ} and define a linear mapping tr : C → ⨁_{ξ∈Ξ} F^{ξ}, where F^{ξ} = F · id_{B^{ξ}} ⊆ End(B^{ξ}), letting

tr(⊕_{ξ}a^{ξ}) = ⊕_{ξ}tr_{ξ}(a^{ξ}) · id_{B^{ξ}} ,

where tr_{ξ}(x) is the trace of the operator R_{p_{ξ}(x)} ∈ End(P(B^{ξ})) and p_{ξ}(x) is the projection of x ∈ C onto P(B^{ξ}). Observe that ⨁_{ξ∈Ξ} F^{ξ} lies in the centroid of C; consider the subalgebra Ψ with respect to this tr(x).

Lemma 1.9. The algebra F(C) is integral over Ψ.

Proof. Let u ∈ F(C); by Cayley-Hamilton’s theorem in Linear Algebra, E_{ξ} = R_{p_{ξ}(u)} satisfies the equation

E_{ξ}^{m+1} − tr_{ξ}(E_{ξ})E_{ξ}^{m} + … + (−1)^{m} det(E_{ξ})E_{ξ} = 0 ,

where m = dim(P(B^{ξ})) and all coeficients are polynomials in tr_{ξ}(E_{ξ}^{j}) [34, p.18]. If we replace each tr_{ξ}(E_{ξ}^{j}) with tr(E^{j}), where E = ⨁_{ξ} E_{ξ}, we get a similar equality for E_{ξ}, thus, E_{ξ} is integral over Ψ; hence, E is also integral over Ψ.

Let f(t) = t^{n} + ∑_{j=1}^{n−1} t^{j}ϕ_{j} be a polynomial over Ψ such that f(E) = 0; put E′ = R_{p(u)}, where R_{x} means the right multiplication operator in C and p(x) = ⨁_{ξ} p_{ξ}(x). Then E′ restricts to E on P(C) = ⨁_{ξ} P(B^{ξ}), hence f(E′) restricts to 0 on P(C) and therefore f(E′) = 0 on C. Notice that u = p(u) + d(u), where d(x) is the projection of x onto the radical of C, hence f(R_{u}) = f(R_{p(u)} + R_{d(u)}) = R_{v}, where v is a mapping in Ψ · F(C) which takes value in the radical R(C) under any specialization of variables in A. Hence, u · f^{c}(R_{u}) = 0 for all c > b(C), therefore u is integral over Ψ. □

Put Υ = S_{a,b,r}(Γ) + T (C), it is precisely the right hand side of (1.22). Next, let I = Υ/T (C) ⊆ F(A); to use proposition 1.8 it suffices to show that I is invariant under multiplication by elements in Ψ.

Lemma 1.10. For all h ∈ Υ and u ∈ F(C) there exist g ∈ Υ such that modulo T (C)

tr(u) · h = g  (1.24)

Proof. We need to prove equality (1.24) under any specialization of variables in C. Since I is a fully characteristic ideal, we may assume that h = S_{a,b,r}(f), where f is a multilinear polynomial in Γ and u is a variable, u ∉ varb(f). To find g we will need the following important observation [29].

Lemma 1.11. Let V, W be vector spaces over a field F, where dim(V) = n. Let f = f(x_{1}, …, x_{n}) be a multilinear skew symmetric mapping from V to W. Then for all E ∈ End(V)

tr(E) · f = ∑_{i=1}^{n} f|_{x_{i}=x_{i}E} .

Proof. It suffices to check it for all x_{1}, …, x_{n} ∈ V; since f is skew symmetric, we can assume that they form a basis of V. Let (a_{ij}) be the matrix of E in this base. Then

f|_{x_{i}=x_{i}E} = f|_{x_{i}=∑_{j} a_{ij}x_{j}} = f|_{x_{i}=a_{ii}x_{i}} = f · a_{ii} .

Taking the sum , we get the required statement. □

Put

g = f ∗ tr(u) = ∑_{y∈Θ_{1}} f|_{y=yu} ,

clearly, it belongs to S_{a,b,r}(Γ). Since the both sides of (1.24) are multilinear, it suffices to check the equality under specialization x → x̄ ∈ Spl(C). Moreover, since C is the direct sum of algebras B^{ξ} and tr(x), respectively, is the direct sum of tr_{ξ}(x), we need only to prove that

f ∗ tr(u) = tr_{ξ}(u) · f

under any substitution x → x̄ ∈ Spl(B^{ξ}). Moreover, thanks to (1.23) we can assume that t(B^{ξ}) = (a, b).

Consider the case when the right hand side does not vanish under x → x̄ (the proof similar under the same hypothesis for the other side). First, observe that ū ∈ P(B^{ξ}); next, a(B^{ξ}) = a, hence for every i ∈ {1, …, b} there exists x ∈ Λ_{i} with x̄ ∈ R(B^{ξ}). Moreover, b(B^{ξ}) = b, hence for all x ∉ ⋃_{i=1}^{b} Λ_{i} the value x̄ belongs to P(B^{ξ}). In particular, x̄ ∈ P(B^{ξ}) for all x ∈ Θ_{1}; besides, |Θ_{1}| = a is equal to a(B^{ξ}), the dimension of P(B^{ξ}), hence the required statement follows from lemma 1.11. □

Thus, proposition 1.6 is proven. Consider the “counter-example” from corollary 1.3; the family consisting of a single algebra B^{ξ} = A satisfies the hypothesis of proposition 1.6, hence T (C) = S_{a,b,r}(Γ) + T (A) for some finite dimensional algebra C, so T (C) ⊂ Γ is a ”counter-example”. From proposition 1.2 it follows that t(Γ, C) < t(Γ, A) which contradicts to the minimality of t(Γ, A), hence, corollary 1.2 is proven.

7. Representability of relatively free algebras

In this section we prove theorem 1.1; suppose that it is untrue, then there exists a Tideal Γ and a finite dimensional algebra A such that T (A) ⊂ Γ and t(Γ, A) is the smallest one for such pairs. Put Γ′ = Γ + T (B), where B is the algebra from proposition 1.6, then there exists a finite dimensional algebra C such that T (C) = Γ′. Indeed,

S_{a,b,r}(Γ′) ⊆ S_{a,b,r}(Γ) ⊆ T (B) ,

hence, by proposition 1.2 we have t(Γ, B) < (a, b) = t(Γ, A), hence t(Γ′, B) < t(Γ, A) and, by the induction hypothesis, T (C) = Γ′ for some finite dimensional algebra C.

Since t(Γ, A) < (a, b + 1), there exists l ∈ ℕ such that S_{a,b+1,l}(Γ) ⊆ T (A). Put d = (a+1)(b+1)+a·l and let Q = K(A, Y, T (A), d); by lemma 1.4, T (A) ⊆ T (Q) and A ⊆ Q, hence T (A) = T (Q). Consider the direct sum of algebras P = Q ⊕ C; notice P(Q) ⊕ P(C) is a maximal semisimple subalgebra, so we denote it by P(P). Let tr_{Q}(x) (tr_{C}(x)) denote the trace of the linear operator R_{p(x)} ∈ End(P(Q)) (End(P(C)), respectively), where p(x) is the projection onto the semisimple part; define a linear mapping tr : P → C(P) letting

tr(x ⊕ y) = tr_{Q}(x) ⊕ tr_{C}(y) ∈ F · id_{P} ⊕ F · id_{C} .

Consider the algebra Ψ ⊆ Pol(P⃗ , C(P)) generated by tr(u), where u ∈ F(P), then, like in lemma 1.9, one can show that F(P) is integral over Ψ. Let I denote the Ψ-submodule generated by Γ̄ = Γ/T (P) ⊆ F(P) in Ψ · F(P).

Lemma 1.12. I ∩ F(P) = Γ̄.

Proof. Obviously, Γ̄ ⊆ I ∩ F(P). Conversely, take a multilinear polynomial f ∈ I ∩ F(P), it belongs to the Ψ-submodule I′ generated by Γ̄′ = Γ′/T (P) ⊆ F(P). Observe that I′ ∩ F(P) = Γ̄′; indeed, Γ̄′ ⊆ I′ ∩ F(P) and, conversely, I′ ∩ F(P) consists of functions which vanish under substitution of elements in C in place of variables, hence I′ ∩ F(P) ⊆ T (C)/T (P) = Γ̄′. So, f = g + h, where g ∈ Γ and

h ∈ T (B) = ⋂_{ξ∈Ξ} T (B^{ξ}) .  (1.25)

On the other hand, f = ∑_{p=1}^{m} f_{i}ϕ_{i}, where f_{i} ∈ Γ̄ and ϕ_{i} ∈ Ψ. Hence,

h = −g + ∑_{i=1}^{m} f_{i}ϕ_{i} .  (1.26)

Since Γ and Ψ are homogeneous, we may assume that all terms are multilinear,

varb(h) = varb(g) = varb(f_{i}) ∪ varb(ϕ_{i})  (1.27)

and varb(f_{i}) ∩ varb(ϕ_{i}) = ∅ for all i ∈ {1, …, m}. We claim that that h ∈ T (A).

On the contrary, suppose that V(h, A) is not empty; if t(h, A) ≤ t(Γ, A), then, by proposition 1.3 h ∉ ⋂_{ξ∈Ξ} T (B^{ξ}) which contradicts (1.25). So, t(h, A) > t(Γ, A) = (a, b), hence, t(h, A) ≥ (a, b + 1). By proposition 1.2, S_{a,b+1,l}(Υ) ⊈ T (A), where Υ = (h)_{⊤}. Take

h′ = S_{a,b+1,l}(f′)

for some multilinear f′ ∈ Υ and let x̄ ∈ V(h′, A).

The order of the set (⋃_{i=1}^{b+1} Λ_{i}) ∪ (⋃_{j=1}^{r} Θ_{j}) is equal to (b + 1)(a + 1) + a · l = |Y |, so we can identify it with Y. Consider a homomorphsim ψ : F ⟨Y ⟩ → A which takes every x ∈ Y to x̄; it extends to a homomorphism π from A ∗ F ⟨Y ⟩ to A with π|_{A} = id_{A}. For each x ∈ varb(g) we define x̃ ∈ A ∗ F ⟨Y ⟩ in the following way

x̃ = \{ \begin{matrix}x & , if x ∈ Y, \\ x̄ & , otherwise . \end{matrix}

Observe that π(h′(x̃)) = h′(π(x̃)) = h′(x̄) ≠ 0; hence, h̃′ = h′(x̃) ≠ 0.

Also, since h′ is multilinear, h̃′ is homogeneous with respect to Y of total degree d, hence h̃′ ∉ V (in notation of lemma 1.4) and therefore h′(x̂) ≠ 0, where x̂ = x̃/V ∈ Q. On the other hand, in the algebra Ψ · F(P) we have

h′ = −g′ + ∑_{n=1}^{m} f′_{n}ϕ′_{n}

where g′, f′_{n} ∈ Γ̄, ϕ′_{n} ∈ Ψ; also, we may assume that g′, f′_{n}, ϕ′_{n} are multilinear and

varb(h′) = varb(g′) = varb(f′_{n}ϕ′_{n})

for each n. Moreover, since h′ is skew-symmetric with respect to all Λ_{i} and Ω_{j}, so is g′ and f′_{n}ϕ′_{n}, for just apply the operator \frac{1}{|G|}S_{G} (see (1.15)) to both sides.

Next, we claim that g′(x̂) = (f′_{n}h′_{n})(x̂) = 0 for all n. Indeed, g′ ∈ S_{a,b+1,r}(Γ) ⊆ T (A) = T (Q); the same holds for f′_{n}, if Y ⊆ varb(f_{n}). So, there is x ∈ Y ∩ varb(ϕ′_{n}), by definition x̂ ∈ R(Q), hence, ϕ̂′_{i} = 0. Thus, h′(x̂) = 0 which is impossible, hence h ∈ T (A)/T (R) ⊆ Γ̄ and therefore f = g + h ∈ Γ̄. Lemma 1.12 is proven. □

By lemma 1.8, Γ̄ = I ∩ F(Q) is equal to T (D)/T (Q), where D is a finite dimensional algebra. Thus, Γ = T (D) and theorem 1.1 is proven. □

24 1. KEMER’S THEOREM

CHAPTER 2 Identities of Lie algebra representations

The rest of these notes is devoted to the problem of finite basis of identities in finite dimensional Lie algebras and their representations over a field of characteristic zero. In this chapter we formulate the theorem and obtain some preliminary results. In particular, we introduce the class of generalized Lie modules, where the theorem corresponds to a more general fact; then we give the notion of “counter-example” and define the type of a “counter-example.

1. Theorem

Let L be a Lie algebra over a field F of characteristic 0 and let ρ : L → End(V) be a Lie representation. An associative polynomial f = f(x_{1}, …, x_{n}) ∈ A⟨X⟩ is said to be an identity of the representation ρ, if f(ρ(a_{1}), …, ρ(a_{n})) = 0 for all a_{i} ∈ L. Obviously, we may assume that ρ is faithful, hence, the definition can be rewritten in terms of Lie pairs as follows (cf. [33, chap.1]).

Let A be an envelope of a Lie algebra L, then P = (A, L) is called a Lie pair; the class of Lie pairs over F is a variety of 2-base algebraic systems. By Poincaré-Birkhoff-Witt’s theorem, A⟨X⟩^{#} is the universal envelope of the free Lie algebra L⟨X⟩, hence (A⟨X⟩^{#}, L⟨X⟩) is a free Lie pair P⟨X⟩. An identity of P is a polynomial f(x_{1}, x_{2}, ..., x_{n}) ∈ A⟨X⟩ such that f(a_{1}, a_{2}, ..., a_{n}) = 0 for all a_{i} ∈ L, where the products are taken in A. The set of all identities of A is an ideal T (P) of A⟨X⟩^{#} closed under replacing variables with Lie polynomials. In other words, T (P) is a fully characteristic ideal of P⟨X⟩; following Rasmyslov’s terminology [33], we call it a weak T-ideal.

The rest is devoted to the proof of the following theorem.

Theorem 2.1. Let P = (A, L) be a Lie pair, where L is a finitely generated Lie algebra and A is an envelope of L satisfying a polynomial identity (i.e., A is a PI algebra). Then, T (P) is generated as a weak T-ideal by a finite number of polynomials.

In terms of Lie representations the theorem says that a PI representation of a finitely generated Lie algebra has a finite basis of identities, cf. [4, p.182].

Recall that the adjoint representation ad : L → End(L) takes an element a to the operator of right multiplication ad(a) : x → [x, a] (we use this notation to distinguish it from the operator of right multiplication R_{a} : x → xa in its envelope).

Corollary 2.1. Let L be a finitely generated Lie algebra such that the envelope of ad(L) in End(L) is PI, then T (L) is a finitely generated T-ideal in the free Lie algebra L⟨X⟩.

Proof. Consider a Lie pair P = (A, ad(L)) where A is the envelope of ad(L) in End(L). Obviously, it satisfies the hypothesis of the theorem, hence, the weak T-ideal T (P) has a finite set of generators f_{1}, f_{2}, ..., f_{n}. Let y ∈ X such that f_{1}, f_{2}, ..., f_{n} do not depend on y, then the set of Lie polynomials

S = { yf_{i}(ad(x_{1}), ad(x_{2}), ..., ad(x_{m})) | f_{i} = f_{i}(x_{1}, x_{2}, ..., x_{m}), i = 1, ..., n }

generates the T-ideal T (L). Indeed, T (L) is generated by multilinear polynomials and we may assume that they depend on y; pick such a polynomial, it can be written in the form, cf. [5, p.71],

y · h(ad(x_{1}), ad(x_{2}), ..., ad(x_{k})) ,

where h is an associative polynomial. Obviously, h is an identity of P, hence we can replace it by E_{1}f_{i}(ad(v_{1}), ad(v_{2}), ..., ad(v_{m}))E_{2}, where v_{i} are Lie polynomials and E_{1}, E_{2} are products of adjoint operators. Obviously,

y · E_{1}f_{i}(ad(v_{1}), ad(v_{2}), ..., ad(v_{m}))E_{2}

belongs to the T-ideal generated by y · f_{i}(ad(x_{1}), ad(x_{2}), ..., ad(x_{m})). □

Since any finite dimensional algebra is PI (it satisfies a Capelli identity), we get immediately an analog of Kemer’s theorem about finite basis of identities in finite dimensional Lie algebras.

Corollary 2.2. The ideal of identities of a finite dimensional Lie algebra is a finitely generated T-ideal.

Since F is infinite, every weak T-ideal is homogeneous by lemma 1.2, hence in the hypothesis of theorem 2.1 we can assume that F is algebraically closed. Indeed, let H be is a field extension of F, then the free associative algebra over H is isomorphic to H ⊗_{F} A⟨X⟩. Consider the scalar extension P_{H} = (A_{H}, L_{H}), where A_{H} = H ⊗_{F} A and L_{H} = H ⊗_{F} L. Then T (P_{H}) = H ⊗_{F} T (P) and T (P_{H}) is a finitely generated weak T-ideal iff so is T (P).

2. Capelli polynomials

Recall that the T-ideal of Capelli identities of rank n, Cap_{n}, is generated by multilinear polynomials that are skew-symmetric with respect to n variables, see (1.4 ).

Proposition 2.1. Cap_{n} is a finitely generated weak T-ideal.

Proof. Denote by Cap_{n,k} the weak T-ideal generated by polynomials

g = f(u_{1}, u_{2}, ..., u_{n}, ...),  (2.1)

where f(x_{1}, x_{2}, ..., x_{n}, ...) is multilinear and skew-symmetric with respect to {x_{1}, x_{2}, ..., x_{n}} and u_{1}, u_{2}, ..., u_{n} are associative polynomials, deg u_{i} ≤ k for all i ∈ {1, …, n}. Then Cap_{n} = ⋃_{k≥1} Cap_{n,k}.

First, let us show that Cap_{n,k} is a finitely generated weak T-ideal for all k; in fact, we follow Rasmyslov’s proof, see [5, p.225]. Obviously, as a weak T-ideal, Cap_{n,k} is generated by polynomials

∑_{σ∈S_{n}} sgn(σ)(u_{σ(1)}E_{1}) … (u_{σ(n−1)}E_{n−1})(u_{σ(n)}E_{n}) ,  (2.2)

where u_{i} are associative monomials, deg(u_{i}) ≤ k, and E_{j} are products of adjoint operators (E_{j} may be equal to 1). We claim that all polynomials of this form, where deg(E_{j}) ≤ n, generate Cap_{n,k} as a weak T-ideal. Notice that their degree is bounded, hence it will yield the required statement.

Let I denote the weak T-ideal generated by polynomials (2.2); clearly, I ⊆ Cap_{n,k}. To prove the reciprocal inclusion consider a tensor product of vector spaces over F

V = A⟨X⟩^{⊗n} ⊗ B^{⊗n}

where B denotes the envelope of ad (L⟨X⟩) in the algebra of endomorphisms of the vector space A⟨X⟩. Also, define a linear mapping g : V → A⟨X⟩^{#} letting

g(u_{1} ⊗ … ⊗ u_{n} ⊗ E_{1} ⊗ … ⊗ E_{n})

be equal to element (2.2), where u_{i}, E_{j} are elements from the corresponding tensor factors of V; obviously, the image g(V) generates Cap_{n,k} as a weak T-ideal.

Denote by W_{m} the span of elements

w = u_{1} ⊗ … ⊗ u_{n} ⊗ E_{1} ⊗ … ⊗ E_{n} ,  (2.3)

where u_{i} is an associative product of ≤ k Lie polynomials and the operator length of E_{j} is ≤ m. By definition, the ideal I is generated by g(W_{n}), and we need to show that g(W_{m}) ⊆ I for all m > n. We shall carry out the induction on m; the initial step, m = n, is trivial, suppose that it holds for m = s, let us prove it for m = s + 1.

Let S denote the algebra of commutative polynomials over F in variables t_{1}, t_{2}, ..., t_{n}. Fix a free generator y ∈ X and define a representation of S in End(V) setting

(u_{1} ⊗ … ⊗ u_{n} ⊗ E_{1} ⊗ … ⊗ E_{n}) ∗ t_{j} =
= (u_{1} ⊗ … ⊗ u_{n} ⊗ E_{1} ⊗ … ⊗ E_{n})|_{E_{j}=ad(y)E_{j}} .

Lemma 2.1. Suppose that ϕ is a symmetric polynomial in S, then g(W_{m}∗ϕ) ⊆ g(W_{m}) for all m ∈ ℕ.

Proof. Take a generator w ∈ W_{m} of form (2.3), then

w ∗ t_{1}^{q_{1}}t_{2}^{q_{2}}...t_{n}^{q_{n}} = w|_{∀i E_{i}=ad(y)^{q_{i}}E_{i}} =
= (u_{1} ⊗ … ⊗ u_{n} ⊗ ad(y)^{q_{1}}E_{1} ⊗ … ⊗ ad(y)^{q_{n}}E_{n}) .

Hence,

g(w ∗ ∑_{τ∈S_{n}} t_{τ(1)}^{q_{1}}t_{τ(2)}^{q_{2}}...t_{τ(n)}^{q_{n}}) = g(w ∗ ∑_{τ∈S_{n}} t_{1}^{q_{τ(1)}}t_{2}^{q_{τ(2)}}...t_{n}^{q_{τ(n)}}) =

= ∑_{σ∈S_{n}} sgn(σ) ∑_{τ∈S_{n}}(u_{σ(1)}ad(y)^{q_{τσ^{−1}σ(1)}}E_{1}) … (u_{σ(n)}ad(y)^{q_{τσ^{−1}σ(n)}}E_{n}) =

= ∑_{τ∈S_{n}} ∑_{σ∈S_{n}} sgn(σ)(u_{σ(1)}ad(y)^{q_{τσ(1)}}E_{1}) … (u_{σ(n)}ad(y)^{q_{τσ(n)}}E_{n}) = ∑_{τ∈§_{n}} f_{τ},

where

f_{τ} = ∑_{σ∈S_{n}} sgn(σ)(v_{σ(1)}^{τ}E_{1}) … (v_{σ(n)}^{τ}E_{n})

with v_{j}^{τ} = u_{j}ad(y)^{τ(j)} for all j. Since v_{j} is a linear combination of associative products of ≤ k Lie polynomials, we have f_{τ} ∈ g(W_{m}) and therefore g(w ∗ ϕ) ∈ g(W_{m}). □

Now take w ∈ W_{s+1} of form (2.3); notice that a transposition of operators in E_{j} reduces the operator length of E_{j}, hence, we may assume that E_{j} = ad(y)^{n}E′_{j} for some y ∈ X. Hence w = w′ ∗ t_{j}^{n}, where w′ is of form (2.3) with E_{j} replaced by E′_{j}, in particular, w′ ∈ W_{s−n+1}.

Next, t_{j} is the root of the polynomial

t^{n} + t^{n−1}ϕ_{1} + … + ϕ_{n−1}t + ϕ_{n} = ∏_{i=1}^{n}(t − t_{i})

where ϕ_{r} are symmetric polynomials. Hence, by the induction hypothesis and the lemma above for all r ≥ 1

g(w′ ∗ (t_{j}^{n−r}ϕ_{r})) = g((w′ ∗ t_{j}^{n−r}) ∗ ϕ_{r}) ∈ g(W_{s+1−r} ∗ ϕ_{r}) ⊆ g(W_{s+1−r}) ⊆ g(W_{s}) ⊆ I

and

g(w) = g(w′ ∗ t_{j}^{n}) = − ∑_{r=1}^{n} g(w′ ∗ (t_{j}^{n−r}ϕ_{r})) ∈ I

as required.

In a similar fashion one can show that Cap_{n} = Cap_{n,n}; actually, Cap_{n} is generated by

∑_{σ∈S_{n}} sgn(σ)v_{0}u_{σ(1)}v_{1} … u_{σ(n−1)}v_{n−1}u_{σ(n)}v_{n} ,

where u_{j}, v_{j} are associative polynomials; we need to prove that Cap_{n} is generated by such polynomials where deg(u_{j}) ≤ n. Suppose that deg(u_{j}) > n, then we can assume that u_{j} = u′_{j}y^{n}. Distributing “uniformly” powers of y on v_{1}, …, v_{n}, we reduce the power of y in u_{j}. □

Lemma 2.2. Theorem 2.1 follows from a.c.c. for weak T-ideals containing the ideal of identities of a finite dimensional Lie pair.

Proof. By [18] there is a natural n such that Cap_{n} ⊆ T (A) ⊆ T (A). Also, from [24] and nilpotency of the radical of a f.g. PI algebra [33, 18, 8] it follows that there exists a f.d. algebra B such that the ideal of identities T (B) is contained in Cap_{n}. Obviously, T (B) is equal to the ideal of identities of the Lie pair B = (B, B^{(−)}). By hypothesis, there are f_{1}, …, f_{m} ∈ T (A) which generate T (A) modulo T (B) as a weak T-ideal. Also, let g_{1}, …, g_{k} be generators of the weak T-ideal Cap_{n}, then f_{1}, …, f_{n}, g_{1}, …, g_{k} generate T (A). □

3. Generalized Lie modules

Generalized Lie representation. The notion of representation of an algebraic system (ring, algebra, group) is related to some action of this system. Thus, a representation μ of a Lie algebra L means that an element a from L acts as a linear operator μ(a) = μ_{a} on a vector space V, and μ is a homomorphism from L to the Lie algebra End(V)^{(−)}. It induces the dual representation ρ^{∗} of L in the endomorphism algebra of the space of linear forms V^{∗}, where for f = f(v) ∈ V^{∗} and a ∈ L we put (f · μ_{a}^{∗})(v) = −f(v · μ_{a}). Moreover, we can replace V^{∗} with the space of linear mappings Hom(V, W) from V to another vector space W.

Now suppose that μ is the adjoint representation ad(x), then the induced representation ρ = μ^{∗} in Hom(L, W) has the form

f · ρ(a) = f|_{x=[a,x]}

In this situation there is some additional type of action of L. Namely, fixing the value of the variable, we get an element from W which can be regarded as a constant function on W,

f · δ(a) = f|_{x=a} .

The resulting element is not in Hom_{F} (L, W) which forces us to consider the sum V = Hom(L, W) + W; clearly, it is direct. We can extend trivially ρ(a), δ(a) to V and get finally the following definition:

f · ρ(a) = \{ \begin{matrix}f|_{x=[a,x]} & ,if f ∈ Hom_{F} (L, W), \\ 0 & ,if f ∈ W ; \end{matrix}  (2.4)

f · δ(a) = \{ \begin{matrix}f|_{x=a} & ,if f ∈ Hom_{F} (L, W), \\ 0 & if f ∈ W. \end{matrix}  (2.5)

One can easily check the following relations: for all a, b ∈ L

ρ(a)δ(b) = δ([ab]) ,  (2.6)
δ(a)ρ(b) = δ(a)δ(b) = 0 .  (2.7)

Consider a more general situation; take the direct sum kL = L ⊕ … ⊕ L and let V be the subspace of Pol(kL, W) spanned by all homogeneous polynomials of degree 0 or 1 with respect to every generic element x_{i} of L. Fix x_{i} and for all a ∈ L define ρ_{i}(a), δ_{i}(a) ∈ End(V) setting for all f ∈ V

f · ρ_{i}(a) = \{ \begin{matrix}f|_{x_{i}=[a,x_{i}]} & ,if deg_{x_{i}} f = 1, \\ 0 & ,if deg_{x_{i}} f = 0, \end{matrix}  (2.8)

f · δ_{i}(a) = \{ \begin{matrix}f|_{x_{i}=a} & ,if deg_{x_{i}} f = 1, \\ 0 & ,if deg_{x_{i}} f = 0. \end{matrix}  (2.9)

We can extend this construction to a relatively free Lie algebra L_{𝔐}⟨X⟩ in the following way. First, enlarge the set of free variables setting X′ = X ∪ {x_{1}, …, x_{k}} and let V be the vector space of Lie polynomials from L_{𝔐}⟨X′⟩ of degree ≤ 1 with respect to each x_{i}; in particular, V includes L_{𝔐}⟨X⟩.

Every element f = f(x_{1}, …, x_{k}) from V can be regarded as a mapping from kL_{𝔐}⟨X⟩ to L_{𝔐}⟨X⟩:

(a_{1}, …, a_{k}) → f(a_{1}, …, a_{k}) ,

hence, (2.8), (2.9) define a similar representation of L_{𝔐}⟨X⟩ in End(V).

We have the following relations between ρ_{i}, δ_{j}. For all a, b ∈ L

[ρ_{i}(a), ρ_{i}(b)] = ρ_{i}([a, b]) ,  (2.10)

i.e. , ρ_{i} is an ordinary Lie algebra representation,

ρ_{i}(a)δ_{i}(b) = δ_{i}([a, b]) , δ_{i}(b)ρ_{i}(a) = 0 , δ_{i}(a)δ_{i}(b) = 0 ;  (2.11)

also, if i ≠ j, then

[ρ_{i}(a), ρ_{j}(b)] = 0 , [ρ_{i}(a), δ_{j}(b)] = 0 , [δ_{i}(a), δ_{j}(b)] = 0.  (2.12)

A generalized Lie (g.Lie) representation of rank n of a Lie algebra L in an associative algebra A (1 ∈ A) is an n-tuple ν = (ν_{1}, …, ν_{n}), where ν_{i} is a pair (ρ_{i}, δ_{i}) of linear mappings from L to A and relations (2.10)-(2.12) hold for all a, b ∈ L.

We have Ker(ρ_{j}) ◁ L, because ρ_{j} is a representation of L; also from (2.11) it follows that Ker(δ_{j}) ◁ L, hence, we have the following statement.

Lemma 2.3. The kernel of a g.Lie representation ν : L → A,

Ker(ν) = ⋂_{j=1}^{s}(Ker(ρ_{j}) ∩ Ker(δ_{j}) ,

is an ideal of L.

Put ν_{i}(L) = ρ_{i}(L) + δ_{i}(L), then by (2.10), (2.11) it is a Lie subalgebra of A^{(−)}, where δ_{i}(L) is a trivial ideal of ν_{i}(L) (its square is equal to 0). Moreover, ν_{i}(L), ν_{j}(L) commute for different i, j, hence

ν(L) = ∑_{j=1}^{s} ν_{j}(L)  (2.13)

is a Lie algebra. The associative subalgebra of A generated by ν(L) and 1 ∈ A is called a generalized (g.) envelope and denoted by En(L, ν).

Fix n ∈ ℕ and a Lie algebra L; a g.Lie representation ν : L → A, ν = (ν_{1}, …, ν_{n}) is said to be universal one, if for all g.Lie representation ν′ : L → A′ there exists an algebra homomorphism ψ : A → A′ such that ν′ is equal to the composition ν and ψ. We may assume that A is generated ν(L) and 1 ∈ A; one can easily check that a universal representation exists and unique up to an isomorphism; the algebra A is called the universal generalized (g.) envelope and denoted by U(L, n).

Generalized Lie modules. It would be natural to give this name to a vector space V with a g.Lie representation of L in End(V). However, in the proof of the theorem we will use it for a slightly different object; namely, we fix several vectors in V and consider them as additional operations. Therefore, a g.Lie module is defined as an ordered set

A = (L; V; ν; v_{1}, …, v_{r})  (2.14)

where L is a Lie algebra, V is a vector space, ν = (ν_{1}, …, ν_{s}) is a g. Lie representation of L in End(V) and v_{1}, …, v_{r} ∈ V are fixed vectors in V; we will use often a shorter notation, e.g. A = (L; V).

Let 𝔐 denote the class of g.Lie modules (2.14) over F, where the class of the g.Lie representation s and the number of the fixed vectors r are fixed; a g. Lie module A = (L; V) ∈ 𝔐 can be regarded as a 2-base algebraic system, where the operations are

1) the operations of the Lie algebra L;

2) ∑_{i=1}^{r} v_{i}E_{i}, where E_{i} are linear combinations of products of ρ_{j}(a), δ_{l}(b), where a, b are Lie polinomials from L⟨X⟩,

satisfying all identities following from the linearity of ρ_{j}, δ_{l} and identities (2.10)-(2.12). Obviously, we may assume that V is generated by v_{1}, …, v_{r} as a module over En(L, ν) ⊆ End(V); also, put En(A) = En(L, ν).

A homomorphism from A = (L; V) to A′ = (L′, V′) is a pair (ϕ, ψ), where ϕ : L → L′ is a Lie algebra homomorphism and ψ is a linear mapping from V to V′ taking v_{j} to v′_{j} and

ψ(v · ρ_{j}(a)) = ψ(v) · ρ_{j}(ϕ(a)) ; ψ(v · δ_{j}(a)) = ψ(v) · δ_{j}(ϕ(a))

for all v ∈ V, a ∈ L and j ∈ {1, …, s}.

A submodule B = (M; W) of A consists of a Lie subalgebra M ⊆ L and a subspace W ⊆ V generated as a module over En(M, ν) ⊆ En(A) by v_{1}, …, v_{n}. Let A_{i} = (L_{i}; V), i ∈ I, be a family of g.Lie modules, then the direct product ∏_{i∈I} A_{i} is equal to (∏_{i∈I} L_{i}; ∏_{i∈I} V_{i}), where the j-th fixed vector is the product of the corresponding vectors from V_{i}. Let X be a set of variables; the free g. Lie module is M⟨X⟩ = (L⟨X⟩; V ⟨X⟩), where L⟨X⟩ is a free Lie algebra and V ⟨X⟩ is a free module over U(L⟨X⟩, s) generated by v_{1}, …, v_{r}.

In fact, we need to modify the notion of g.Lie module: we have to replace an ordinary Lie algebra with k-sorted one.

Generalized Lie modules over k-sorted Lie algebras.

Let L be a Lie algebra which is a sum of fixed ideals L_{1}, …, L_{k} ◁ L, then L is called a k-sorted Lie algebra. Formally speaking, it is a k + 1-tuple L = (L; L_{1}, …, L_{k}); we consider it as k + 1-base algebraic system, cf. [33], where all operations are defined on L_{i_{1}} × … × L_{i_{r}}, where i_{1}, …, i_{r} ∈ {1, …, k} and take value either in some L_{i} or in L. Operations of the first type include all Lie algebra operations on L_{i} and action of L_{j} on L_{i} by derivations; operations of the second type are defined by Lie polynomials in many variables. One can write down identities which enable us to identify each L_{i} with an ideal of L; so the class of k-sorted Lie algebras over F form a variety of algebraic systems. In case k = 1 we get ordinary Lie algebras.

The expression x ∈ L will mean that x belongs to one of L_{i}.

Let L, L′ be k-sorted Lie algebras; a homomorphism from L to L′ is Lie algebra homomorphism π : L → L′ taking every L_{i} to L′_{i}. A k-sorted subalgebra of L is a subalgebra M of L which is the sum of ideals M_{i} ⊆ M ∩ L_{i}, i ∈ {1, …, k}. Take a k-sorted subset S = ∪_{i=1}^{k}S_{i}, S_{i} ⊆ L_{i}; then the k-sorted subalgebra generated by S is equal to (M; M_{1}, …, M_{k}), where M is a subalgebra of L generated by S and M_{i} is the ideal of M generated by S_{i}.

Let X be a disjoint union of infinite sets of variables, X = ⋃_{i=1}^{k} X_{i}, and let L_{i}⟨X⟩ denote the ideal of L⟨X⟩ generated by X_{i}. Then it defines a k-sorting on L⟨X⟩; moreover, every mapping π : X → L, which takes X_{i} to L_{i}, extends to a homomorphism of k-sorted Lie algebras. Thus, (L⟨X⟩; L_{1}⟨X⟩, …, L_{k}⟨X⟩) is the free k-sorted Lie algebra L⟨X⟩.

A generalized Lie representation of rank s of a k-sorted Lie algebra L = (L; L_{1}, …, L_{k}) in an associative algebra A (1 ∈ A) is that of L; put En(L, ν) = En(L, ν). Finally, a generalized Lie module A is

(L; V) = (L; V; ν; v_{1}, …, v_{r})

where L is a k-sorted Lie algebra, V is a vector space, ν is a g.Lie representation of L in End(V) and v_{1}, …, v_{r} are fixed vectors from V.

Cleraly, we can replace the ground field F with an arbitrary associative commutative ring K and get g.Lie modules over K.

Like in the case of ordinary Lie algebras (i.e., 1-sorted Lie algebras) we consider a g.Lie module as an algebraic system; so we assume that V is generated by v_{1}, …, v_{r} over En(A) = En(L, ν) ⊆ End(V).

Let 𝔐_{k,s,r} denote the class of g.Lie modules over F with k, s, r fixed; clearly, it is a variety and all basic algebraic constructions extend automatically to this case. In partucular, the free g.Lie module is

M⟨X⟩ = (L⟨X⟩; V ⟨X⟩) .  (2.15)

where V ⟨X⟩ is a free module over U(L⟨X⟩, s) = U(L⟨X⟩, s) generated by v_{1}, …, v_{r}.

Clearly, U(L⟨X⟩; s) and, therefore, V ⟨X⟩ are homogeneous with respect to every x ∈ X, so deg_{x}(f), deg_{x}(E) are defined for all f ∈ v⟨X⟩, E ∈ U(L⟨X⟩; s) which are homogeneous with respect to x.

Let A be a g.Lie module from 𝔐_{k,s,r}; a polynomial f = f(x_{1}, …, x_{n}) ∈ V ⟨X⟩ is said to be an identity of A, if it vanishes under any homomorphism from M⟨X⟩ to A. In other words, f is equal to 0 ∈ V under any substitutions of elements from L in place of variables (note that every variable from X_{i} must be replaced with an element from L_{i}).

Let T (A) denote the set of all identities of A; it is a submodule of V ⟨X⟩ over En(M⟨X⟩) closed under substitutions of Lie polynomials g ∈ L_{i}⟨X⟩ in place of x ∈ X_{i}. In other words, T (A) is a fully characteristic submodule of V ⟨X⟩, i.e., it is invariant under any g.Lie module endomorphism of M⟨X⟩. We prefer to call such a submodule a T-ideal like in the case of algebras.

Since F is infinite, every T-ideal Γ is homogeneous with respect to every variable x ∈ X; it means that Γ is decomposed into the direct sum of its homogeneous components. It follows from lemma 1.2; recall that the proof is based on the Vandermonde determinant argument employing substitutions of the form x → λx, where λ ∈ F, hence, the statement can be rewritten as follows. Let f ∈ V ⟨X⟩ and let g be its homogeneous component with respect to some variables, then there exist endomorphisms (ϕ_{1}, ψ_{1}), …, (ϕ_{l}, ψ_{q}) of M⟨X⟩ such that g = ψ(f), where ψ is a linear combination of ψ_{1}, …, ψ_{q}. In other words, the projector onto a homogeneous component of V ⟨X⟩ acts on a given element like a linear combination of endomorphisms of M⟨X⟩, however, this representation is local, it depends on the degree of the element.

Let E(X) ⊆ End(V ⟨X⟩) denote the subalgebra generated by the following elements:

(1) endomorphisms of M⟨X⟩;

(2) projectors onto homogeneous components of V ⟨X⟩ with respect to Y ⊆ X. So, a submodule of V ⟨X⟩ is a T-ideal iff it is invariant under E(X).

Recall that V ⟨X⟩ is the free module over En(M⟨X⟩) = U(L⟨X⟩, s), hence, every endomorphism ϕ : L⟨X⟩ → L⟨X⟩ extends to an endomorphism ψ′ of U(L⟨X⟩, s) and therefore defines an endomorphism (ϕ, ψ) of M⟨X⟩: ψ(∑_{j=1}^{r} v_{j} · E_{j}) = ∑_{j=1}^{r} v_{j} · ψ′(E_{j}) . Clearly, the correspondence between ψ and ψ′ is one-to-one, so it does not make any confusion, if we denote ψ′ by ψ. So, if f = ∑_{j=1}^{r} v_{j} · E_{j} is an arbitrary element in V ⟨X⟩, then the action of ψ can be rewritten as

ψ(f) = ∑_{j=1}^{r} v_{j} · ψ(E_{j}) .  (2.16)

Also, the same holds for projectors onto homogeneous components; moreover, it extends to the linear combinations and, in particular, to all ψ ∈ E(X).

If I is a T-ideal, then it is equal to the ideal of identities T (A), where A is the quotient (L⟨X⟩, V ⟨X⟩/I) . Let A = (L; V) be a g.Lie module, then elements of F(L; V) = V ⟨X⟩/T (A) can be regarded as functions from L_{1} × … × L_{k} to V in many variables.

Next, consider the algebra of all mappings from L_{1} × … × L_{k} to L in many variables; a mapping from L_{i} (for some i) to L that takes a ∈ L_{i} to a ∈ L is called a generic element of L (of type i). Let X_{i} denote an infinite set of generic elements of type i; also, put X = ⋃_{i=1}^{k} X_{i} and let F(L) denote the subalgebra generated by X. It is equal to the sum of ideals F(L)_{i} generated by X_{i}, so

F(L) = (F(L); F(L)_{1}, …, F(L)_{k})  (2.17)

is a k-sorted Lie algebra called the algebra of generic elements of L.

The g.Lie representation of L in End(V) induces that of F(L) in End(F(V, L)); it defines a g.Lie module

F(A) = (F(L); F(L; V))  (2.18)

called the g.Lie module of generic elements of A.

Now we are going to state a theorem about T-ideals in 𝔐_{k,s,r} which yields theorem 2.1. In fact, we consider a subclass of g.Lie modules satisfying some identities allowing us to rearrange elements in ν(L_{i}) and ν(L_{j}) for i ≠ j.

Let J be the ideal of En(A) generated by Lie commutators [ν(L_{i}), ν(L_{j})], ∀i, j ∈ {1, …, k}, i ≠ j, then

V · J^{m} = 0 .  (2.19)

for some m ∈ ℕ; clearly, it is equivalent to a set of identities. By (2.10)-(2.12), [L_{i}, L_{j}]^{m} ⊆ Ker(ν); so, if ν is faithful, then

[L_{i}, L_{j}]^{m} = 0 .  (2.20)

A g.Lie module A = (L; V) is said to be finite dimensional (f.d) if the dimension of L_{1}, …, L_{k} and V is finite. Suppose L satisfies (2.20), then [L_{i}, L_{j}] is contained in the solvable radical of [L, L] denoted by N(L); from (2.10)-(2.12) it follows that the image of every solvable ideal is a solvable ideal of the Lie algebra ν(L), hence, ν(N(L)) ⊆ N(ν(L)). Thanks to ([17, p.50,56]) N(ν(L) generates a nilpotent ideal in En(A), so (2.19) follows from (2.20) in this case. Hereafter every f.d. k-sorted Lie algebra is supposed to satisfy (2.20) for all i ≠ j.

Let A = (L, V) be a g.Lie module; denote by Q(A) the T-ideal generated by all multilinear f ∈ V ⟨X⟩ satisfying the following condition: for all i, j ∈ {1, …, k}, i ≠ j, and x ∈ X_{i}, y ∈ X_{j}, where deg_{x}(f) ≠ 0,

f|_{x=[x,y]} ∈ T (A) .  (2.21)

In other words, f vanishes under substitution any elements in L in place of variables, where x is replaced by an element from ∑_{j≠i}[L_{i}, L_{j}]. Note that in case k = 1 this condition becomes trivial, so we put Q(A) = V ⟨X⟩.

Theorem 2.2. Let A be a finite dimensional g.Lie module from 𝔐_{k,s,r} satisfying (2.19). Then any ascending chain of T-ideals

T (A) ⊆ Γ_{1} ⊆ … ⊆ Γ_{q} ⊆ … ⊂ Q(A)  (2.22)

becomes stationary.

Let us show that it yields theorem 2.1. Actually, by lemma 2.2 it suffices to show that any ascending chain of weak T-ideals

T (B) ⊆ I_{1} ⊆ … ⊆ I_{q} ⊆ …  (2.23)

where B = (B, M) is a finite dimensional Lie pair, becomes stationary.

Define a covariant functor F from the category of Lie pairs to the category of g.Lie modules 𝔐_{1,1,1}. Given Lie pair A = (A, L) put F(A) = (L; A; (ρ_{1}, δ_{1}); v_{1}), where v_{1} is the identity element of A , ρ_{1}(x) is the operator of right multiplication R_{x} ∈ End(A) and δ_{1} is equal to 0 identically. One can check that any homomorphism f = (ϕ, ψ) between two Lie pairs A and B induces a homomorphism F(f) and this mapping preserves the composition. Moreover, since v_{1} is the identity element, every homomorphism from F(A) to F(B) is induced by a homomorphism from A to B and this correspondence is one-to-one.

Let (id, ψ) be the homomorphism of the free g.Lie module M⟨X⟩ = (L⟨X⟩, V ⟨X⟩) to F(P⟨X⟩) induced by the identity mapping of X. Every endomorphism (π, ϕ) of M⟨X⟩ induces an endomorphism ϕ′ of P⟨X⟩ such that ψ(ϕ(f)) = ϕ′(ψ(f)) for all f ∈ V ⟨X⟩; hence, the full pre-image of (2.23),

ψ^{−1}(T (B)) ⊆ ψ^{−1}(I_{1}) ⊆ … ⊆ ψ^{−1}(I_{q}) ⊆ , …  (2.24)

consists of T-ideals of M⟨X⟩.

Moreover, ψ^{−1}(T (B)) is equal to the ideal of identities of F(B). Actually, let (π, μ) be a homomorphism from M⟨X⟩ to F(B), then the algebra homomorphism π : L⟨X⟩ → M defines a homomorphism of Lie pairs (π, τ) : P⟨X⟩ → B and, therefore, a homomorphism of g.Lie modules (π, ν) : F(P⟨X⟩) → F(B), where μ is equal to the composition of ψ and ν. Hence, ψ^{−1}(T (B)) ⊆ T (F(B)). Conversely, the composition of (id, ψ) with any homomorphism from F(P⟨X⟩) to F(B) must take elements from T (F(B)) to 0, hence ψ^{−1}(T (B)) = T (F(B)).

By theorem 2.2 the chain (2.24) becomes stationary; since ψ is epic, the original chain (2.23) has the same property. □

4. Canonical decomposition

The Kemer’s theorem is based on the splitting theorem for a finite dimensional associative algebra A. Roughly speaking, two numbers, the dimension of the semisimple part P(A) and the nilpotency index of the radical R(A), characterizes a “counter-example” T (A) ⊂ Γ.

A similar situation should be expected in other classes of algebras, where a suitable decomposition exists. Actually, Kushkulei-Rasmyslov’ theorem [23] says that identities distinguish non-isomorphic simple algebras (algebraic systems with multilinear operations) over an algebraically closed field of characteristic 0. Thus, we need to decompose properly a finite dimensional Lie algebra into ”simple” components.

Levi’s theorem provides a decomposition of a finite dimensional algebra L into the direct sum a semisimple algebra D and the solvable radical S(L) [17, p.91]. However, S(L) cannot play the same role as the nilpotent radical in associative algebras, because its action on finite dimensional modules might be not nilpotent. We can replace S(L) with the radical of the commutator subalgebra, N(L) = [S(L), L], therefore we need to find a ”suitable” complement to S(L).

Besides, in the associative case a finite dimensional irreducible modules over a semisimple algebra are embedded into the regular one, so the structure of the radical as a module over the semisimple part is not essential for the proof. In the case of Lie algebras the situation is absolutely different, a simple algebra has a wide range of non-isomorphic finite dimensional irreducible modules.

Let L = (L, L_{1}, …, L_{k}) be a k-sorted finite dimensional Lie algebra over F. Put S_{i} = S(L_{i}) = S(L) ∩ L_{i}, hence S(L) = ∑_{i=1}^{k} S_{i}. A maximal semisimple subalgebra of L is defined up to an automorphism, let D = D(L) denote one of them. Also, denote by D_{i} = D_{i}(L) a semisimple part of L_{i}, D(L_{i}); by [17, p.92] there exists an inner automorphism ϕ_{i} of L taking D_{i} to D; since L_{i} is invariant under ϕ_{i}, we may assume that D_{i} ⊆ D. Hence, D = D(L) is the direct sum of D_{i},

D = ⨁_{i=1}^{k} D_{i} .  (2.25)

Also, put N = N(L); if ρ : L → End(V) is a finite dimensional representation, then the image ρ(N) generates a nilpotent ideal of the envelope of ρ(L) [17, p.50,56]; in particular, N is a nilpotent ideal.

Next, let N_{i} denote

N_{i}(L) = N(L_{i}) + I_{i} ,  (2.26)

where

I_{i} = I_{i}(L) = ∑_{j≠i}[L_{i}, L_{j}] ,  (2.27)

then, N = ∑_{i=1}^{k} N_{i}. Indeed, N_{i} ⊆ N for all i by (2.20) and conversely,

N = [L, S] ⊆ ∑_{i=1}^{k}[L_{i}, S] ⊆ ∑_{i=1}^{k}[L_{i}, S(L_{i})] + I_{i} = ∑_{i=1}^{k} N_{i} .

Since D is semisimple, we can find a D-submodule C_{i} completing N_{i} to S_{i} [17, p.79], hence [C_{i}, D] = 0.

Lemma 2.4. For all i there is a complement C_{i} such that [C_{i}, D] = 0 and the Lie subalgebra ⟨C⟩, where C = ∑_{i=1}^{k} C_{i}, is nilpotent.

Proof. Let c_{1}, …, c_{n} be the union of bases of C_{i}; hence they generate the vector space C. We are going to repalce c_{1}, …, c_{n} by suitable elements c′_{1}, …, c′_{n} ∈ ⟨C⟩ such that c_{j} − c′_{j} ∈ (c_{j}, C) where (c_{j}, C) = ∑_{t≥1} c_{j} · ad(C)^{t}. Using the induction on p, let us prove the following statement.

There exist c_{1}^{(p)}, …, c_{n}^{(p)} ∈ ⟨C⟩ and k_{p} ∈ ℕ such that

(*) for all j ∈ {1, …, n} the element c_{j} − c_{j}^{(p)} belongs to (c_{j}, C); (**) let M_{p} denote the subalgebra ⟨c_{1}^{(p)}, …, c_{n}^{(p)}⟩, then for all j ≤ p and c ∈ [M_{p}, M_{p}]

M_{p} · ad(c_{j}^{(p)} + c)^{k_{p}} = 0 .

In fact, we can start with p = 0 assuming that M_{0} = ⟨C⟩ and the statement is trivial in this case. Next, suppose it holds for p ≥ 0, let us prove it for p + 1.

Since M_{p} is finite dimensional, there exists a polynomial f(x) = g(x) · x^{k} + x^{k}, where k > 0 and g(x) is a polynomial without the absolute term, such that M_{p} ·f(ad(c_{p+1}^{(p)})) = 0. For all j ∈ {1, 2, …, n} put b_{j} = c_{j}^{(p)} · g(ad(c_{p+1}^{(p)})); notice that

b_{j} ∈ (c_{j}^{(p)}, C) ⊆ (c_{j}, C)

for all j and b_{p+1} = 0. Put c_{j}^{(p+1)} = c_{j}^{(p)} + b_{j}, hence c_{p+1}^{(p+1)} = c_{p+1}^{(p)} and

c_{j}^{(p+1)} · ad(c_{p+1}^{(p+1)})^{k} = (c_{j}^{(p)} + b_{j}) · ad(c_{p+1}^{(p+1)})^{k} = c_{j}^{(p)} · f(ad(c_{p+1}^{(p)})) = 0 .

In addition, by (**) for all j ∈ {1, 2, …, p} we have

M_{p} · ad(c_{j}^{(p+1)})^{k_{p}} = M_{p} · ad(c_{j}^{(p)} + b_{j})^{k_{p}} = 0.

The subalgebra M_{p+1} = ⟨c_{1}^{(p+1)}, …, c_{n}^{(p+1)}⟩ is finite dimensional, hence it is spanned by a finite number of commutators in the generators (including the generators). Since ad(x) is a derivation, there exists l ∈ N such that M_{p+1} · ad(c_{p+1}^{(p+1)})^{l} = 0. Let m ∈ ℕ be the nilpotency index of the commutator subalgebra [M_{p+1}, M_{p+1}]. Then one can check that for all c ∈ [M_{p+1}, M_{p+1}]

M_{p+1} · ad(c_{p+1}^{(p+1)} + c)^{l·m+1} = 0 .

Put k_{p+1} to be the maximum of k_{p} and l · m + 1, therefore the statement holds for p + 1.

So, if p = n, then there exist c′_{1} = c_{1}^{(n)}, …, c′_{n} = c_{n}^{(n)} such that c′_{j} − c_{j} ∈ (c_{j}, C) ⊆ N(L) and for every c from the commutator subalgebra of M_{n} = ⟨c′_{1}, …, c′_{n}⟩ the operator ad(c′_{i} + c) acts nilpotently on M_{n}. Hence, the same holds if we replace c′_{i} with a linear combination of c′_{1}, …, c′_{n}, hence M_{n} is a nil algebra and by Engel’s theorem [17, p.3] it is nilpotent. Clearly, the span of {c′_{1}, …, c′_{n}} ∩ L_{i} completes N_{i} to S_{i}. □

Let

C_{i} = C_{i}(L)  (2.28)

denote a complement of N_{i} to S_{i} such that [C_{i}, D] = 0 and the subalgebra ⟨C⟩, where C = ∑_{i=1}^{k} C_{i}, is nilpotent. Then L = D + C + N and ⟨C⟩ commutes with D.

Now we are going to split the ideal N_{i} into a sum of “simple” components. First of all, let N̄_{i} be a complement of N_{i}^{2} + I_{i} to N_{i}; we can regard it as the L-module N/N^{2}; it turns out that all necessary information about l-module structure of N_{i} is contained in N̄_{i}. By complete reducibility of modules over D, we may assume that N̄ is a D-submodule of N_{i}. Also, modulo N_{i}^{2} + I_{i} the ideal N acts trivially on N̄_{i}, so we need only to describe N̄_{i} with respect to the adjoint action of ⟨C⟩.

The ⟨C⟩-module N_{i} is the direct sum of weight subspaces ⨁_{λ∈Λ_{i}} N_{i}(λ) [17, p.41]. For a given subspace M ⊆ N_{i}(λ) put

M^{{0}} = M ;

M^{{l}} = ∑_{c_{1},... ,c_{l}∈C} M · (ad(c_{1}) − λ(c_{1})) … (ad(c_{l}) − λ(c_{l})) ;

M^{[k]} = ∑_{l≥k} M^{{l}} .

Recall that D commutes with elements in C, hence, if M is a D-module, then M^{{k}} is a D-submodule too. In addition, if M is an irreducible D-module and

M_{1} = M · (ad(c_{1}) − λ(c_{1})) … (ad(c_{l}) − λ(c_{l}))  (2.29)

is not equal to 0, then M_{1} and M are isomorphic as D-modules.

Let M be a ⟨C⟩-submodule of N_{i}, then M is the direct sum of weight subspaces M(λ) = M ∩ N_{i}(λ) and

N/M = ⨁_{λ∈Λ_{i}}(N_{i}(λ) + M)/M ≃ ⨁_{λ∈Λ} N_{i}(λ)/M(λ) .  (2.30)

Also, put

M^{{m}} = ∑_{λ∈Λ_{i}} M(λ)^{{m}} , M^{[m]} = ∑_{l≥m} M^{{l}}

for all m ≥ 0.

Now let

M_{i} = M_{i}(L) = N_{i}^{2} + N_{i}^{[1]} + ⟨C_{i}⟩^{2} + I_{i} ;  (2.31)

clearly, it is an ideal of L. By (2.30), N_{i}/M_{i} is the direct sum of N_{i}(λ)/M_{i}(λ); observe that for all c ∈ C the operator ad(c) acts as the scalar λ(c), moreover λ(C_{j}) = 0 for all j ≠ i, because [N_{i}, C_{j}] ⊆ I_{i} ⊆ M_{i}.

Choose a D-module W_{i}(λ) completing M_{i}(λ) to N_{i}(λ) and put

W_{i} = W_{i}(L) = ∑_{λ∈Λ_{i}} W_{i}(λ) ,  (2.32)

then the sum is direct and W_{i} completes M_{i} to N_{i}. Besides,

[D_{j}, W_{i}] = 0  (2.33)

for all j ≠ i. Fix a decomposition of W_{i}(λ), λ ∈ Λ_{i}, into a direct sum of irreducible Dsubmodules; then W_{i}(λ) ≃ W_{i}(λ)+M_{i}/M_{i}, it equips W_{i}(λ) with structure of an L-module. Put

pr(W_{i}) = ⋃_{λ∈Λ_{i}} pr (W_{i}(λ))  (2.34)

so W_{i} is equal to the direct sum of all W_{i,α}, α ∈ pr(W_{i}).

Lemma 2.5. The L_{i}-module W_{i} has no trivial submodules (i.e., nonzero submodules with the trivial action of L_{i}).

Proof. Suppose there is a ∈ N_{i}\M_{i} such that [a, L_{i}] ⊆ M_{i}, then [a, L] ⊆ M_{i}. We may assume that a ∈ W_{i}(0), hence the subspace F a = vect_{F} {a} is a trivial D-module. Let U denote a D-submodule of N_{i} which contains M_{i} and completes F a to N_{i}; since a ∈ W_{i}(0), it contains all W_{i}(λ), λ ≠ 0, hence, [U, C] ⊆ U. Thus, L_{i} = D_{i} + C_{i} + F a + U and

[L_{i}, L] = [D_{i} + C_{i} + F a + U, D_{i} + C_{i} + F a + U] + I_{i} ⊆

⊆ [D_{i}, D_{i}] + [D_{i}, U] + [C_{i}, C_{i}] + [C_{i}, U] + [a, U] + [U, U] + I_{i} ⊆ D_{i} + U .

Hence, a ∉ [L_{i}, L] ⊇ N_{i}, which contradicts to the choice of a. □

Corollary 2.3. The sum W = ∑_{i=1}^{k} W_{i} is direct.

Proof. Pick i and let W = W_{i} ∩ ∑_{j≠i} W_{j}, by definition, L_{i} acts trivially on W_{j}, hence, W is a trivial L_{i}-module and by the lemma above, W = 0. □

Put pr(W) = ⋃_{i=1}^{k} pr(W_{i}), then W is the direct sum of L-modules W_{β}, β ∈ pr(W); also, set W^{[l]} = ∑_{i=1}^{k} W_{i}^{[l]} for all l ≥ 0.

Next, let Gn_{i}(L) = D_{i} + C_{i} + W_{i}, it is a D-module of L_{i}; we also regard it as an L-module L_{i}/M_{i}. It turns out, ⋃_{i=1}^{k} Gn_{i}(L) contains all necessary information about Lmodule structure of N.

Lemma 2.6. For all i ∈ {1, …, k} the Lie algebra L_{i} is generated modulo I_{i} by Gn_{i}(L). Moreover,

N_{i} = ⟨C_{i}⟩^{2} + ∑_{t≥1}[\underbrace{W_{i}^{[0]}, W_{i}^{[0]}, …, W_{i}^{[0]}}_{t}] + I_{i} .  (2.35)

Proof. Obviously, we only need to check (2.35). First, since N_{i} is nilpotent, modulo I_{i} it is generated as a Lie subalgebra by a complement N̄_{i} of N_{i}^{2} + I_{i} to N_{i}. Actually, if N_{i}^{c} = 0, then modulo I_{i}

N_{i} ⊆ N̄_{i} + N^{2} ⊆⊆ N̄_{i} + (N̄_{i} + N_{i}^{2})^{2} ⊆
⊆ N̄_{i} + N̄_{i}^{2} + N_{i}^{3} ⊆ … ⊆ N̄_{i} + N̄_{i}^{2} + N̄_{i}^{3} + … N̄_{i}^{c−1} .

Observe that I = N_{i}^{2} + ⟨C_{i}⟩^{2} + I_{i} is a submodule over ⟨C⟩; let us show that modulo I for all λ

N_{i}(λ) ⊆ W_{i}(λ)^{[0]} .

Since λ is a weight of C, there exists l such that N_{i}^{[l]}(λ) = 0, hence, by (2.31), modulo I

N_{i}(λ) ⊆ W_{i}(λ) + N_{i}(λ)^{[1]} ⊆ W_{i}(λ) + N_{i}(λ)^{{1}} + N_{i}(λ)^{[2]} ⊆
⊆ W_{i}(λ) + (W_{i}(λ) + N_{i}(λ)^{[1]})^{{1}} + N_{i}(λ)^{[2]} ⊆

⊆ W_{i}(λ) + W_{i}(λ)^{{1}} + N_{i}(λ)^{[2]} ⊆ … ⊆ ∑_{m=0}^{l−1} W_{i}(λ)^{{m}} + N_{i}^{[l]}(λ) = W_{i}^{[0]}(λ).

So, N̄_{i} ⊆ W_{i}^{[0]} + ⟨C_{i}⟩^{2}, hence by the previous statement, the right hand side generates the algebra N_{i} modulo I_{i}. □

So, fixing Gn_{i}(L) for all i, we define a decomposition of L called a canonical one. Recall that we study T-ideals lying in Q(A), where A = (L; V) is a finite dimensional

g.Lie module satisfying (2.20). Fix a canonical decomposition of L and consider the k-sorted subalgebra M = (M; M_{1}, …, M_{k}) generated by

Gn(L) = ⋃_{i=1}^{k} Gn_{i}(L) .

By (2.21) the value of any f ∈ Q(A) in A is defined by elements from complements of I_{i} to L_{i}. Hence, by the lemma above, we can replace A with the submodule generated by M,

(M; W)

where W is the submodule over En(M, ν) ⊆ En(A) generated by v_{1}, …, v_{r}.

Lemma 2.7. For all i the vector space M_{i} is equal to the direct sum of D_{i}, C_{i}, and

(C_{i}, C) + ∑_{t≥0}[W_{i}^{[0]}, \underbrace{W^{[0]}, …, W^{[0]}}_{t}] ,  (2.36)

and (C_{i}, C) denotes the ideal of ⟨C⟩ generated by [C_{i}, C].

Moreover, M has a canonical decomposition such that Gn_{i}(M) = Gn_{i}(L) for all i.

Proof. Let M′_{i} denote the sum of D_{i} + C_{i} and the right hand side of (2.36), then M′_{i} ⊆ M_{i}; moreover, M′_{1} + … + M′_{k} is a subalgebra of M containing the set of generators Gn(L), hence M = M_{1} + … + M_{k}. Next, M′_{i} is closed under multiplication by elements in Gn(L), hence it is an ideal of M, also Gn_{i}(L) ⊆ M′_{i} , hence M′_{i} = M_{i}.

The subalgebra D_{i} ⊆ M_{i} ⊆ L_{i} is a maximal semisimple one in M_{i}, hence, we can put D_{i}(M) = D_{i}. Next, the solvable radical S(M_{i}) is equal to the sum of C_{i} and (2.36); by definition,

N_{i}(M) = N(M_{i}) + I_{i}(M) ,

from lemma 2.5 it follows that all W_{i,β} lie in N_{i}(M), hence, N_{i}(M) coincides with (2.36). On the other hand, N_{i}(M) ⊆ N_{i}, hence, C_{i} completes N_{i}(M) to S(M_{i}), so we can define C_{i}(M) as C_{i}. Finally, by definition

M_{i}(M) = N_{i}(M)^{[1]} + ⟨C_{i}(M)⟩^{2} + N_{i}(M)^{2} + ∑_{j≠i}[M_{i}, M_{j}] ;

as C_{i}(M) = C_{i}, it is equal to

W_{i}^{[1]} + (C_{i}, C) + ∑_{t≥1}[W_{i}^{[0]}, \underbrace{W^{[0]}, …, W^{[0]}}_{t}] ,

hence, M_{i}(M) + W_{i} = N_{i}(M). On the other hand, M_{i}(M) ⊆ M_{i}, hence we can put W_{i}(M) = W_{i}. □

5. Cayley-Hamilton polynomial

The underlying idea of Kemer’s theorem can be described as follows. Let Γ be a Tideal containing the ideal of identities of a finite dimensional algebra A, we fit gradually a finite dimensional algebra B such that Γ = T (B). At some point we build a T-ideal Γ′ of a special form related to the structure of A such that the quotient C of the free algebra modulo Γ′ admits an extension by a commutative algebra of trace polynomials K. It is important that K has a balanced size. It is not too large so that Γ′ is invariant under K modulo identities of A and, therefore, the extension of C by K is possible. On the other hand, it is large enough to provide certain finiteness conditions. This section gives an analog of K in the case of Lie algebras.

Let L = (L; L_{1}, …, L_{k}) be a finite dimensional k-sorted Lie algebra over F; if V is a finite dimensional Lie module over L, then the corresponding representation of L in End(V) is denoted by ad_{V} (in fact, we will mostly deal with representations of L induced by the adjoint one).

Recall that F(L) denotes the Lie algebra of generic elements of L; since L is finite dimensional, it lies in the algebra of polynomial mappings in many variables from L_{1}, …, L_{k} to L,

Pol(L⃗, L) ≃ F [L⃗] ⊗_{F} L ,

where F [L⃗] = ∏_{i=1}^{k} F [L⃗_{i}].

Let UF(L) denote the universal envelope of the Lie algebra F(L); it consists of associative polynomials in generic elements of L. If V is an L-module, then for all i the representation ad_{V} : L → End(V) restricts to a representation of L_{i} and therefore induces a homomorphism π_{V} from UF(L) to Pol(L⃗, End(V)) that takes f ∈ F(L) to ad_{V} (f); so the image of E ∈ UF(L) is obtained from E by replacing the generic elements with the corresponding adjoint operators.

We will consider modules over a fixed algebra, so it makes sence to abbrevate notations and put Pol(V) = Pol(L⃗, End(V)); also let Π(V) denote the image of UF(L) in Pol(V). Let K be a subalgebra of F [L⃗] ≃ F [L⃗] · id_{V} ⊆ Pol(V);

An element E ∈ Π(V) is said to be integral over K, if there exist s_{1}, …, s_{n−1} ∈ K such that

E^{n} + s_{1} · E^{n−1} + … + s_{n−1} · E = 0 ;

we call f_{E}(t) = t^{n} + s_{1} · t^{n−1} + … + s_{n−1} · t ∈ K[t] a Cayley-Hamilton (C-H) polynomial for E. If it holds for all E ∈ Π(V), then we say that Π(V) is integral over K. For example, let K = tr_{V} [L] be the subalgebra of F [L⃗] generated by polynomials tr(E) where E ∈ Π(V). Then Π(V) is integral over K of degree dim(V), where f_{E}(t) = g(t)t and g(t) is the standard Cayley-Hamilton polynomial for a linear operator.

Let K · Π(V) be the module over K generated by Π(V); it is an algebra over K. As V is finite dimensional, it is an associative PI algebra, hence, the integrality of Π(V) over a subalgebra K ⊆ F [L⃗] has an equivalent form given by Shirshov’s theorem about height [34, 37].

Lemma 2.8. Π(V) is integral over K iff any finitely generated subalgebra of K ·Π(V) is a finitely generated module over K.

Now our task is to find a “minimal” V such that Π(L) is integral over K = tr_{V} [L]. Fix an L-module V and consider the class Q(V) of all finite dimensional L-modules U such that Π(U) is integral over K = tr_{V} (L).

Lemma 2.9. The class Q is closed with respect to

(i) submodules, (ii) quotients, (iii) tensor products, (iv) finite direct sums (with the diagonal action of L),

(v) extensions: if an L-module V has a chain of submodules

V = V_{1} ⊇ V_{2} ⊇ … ⊇ V_{m} = 0  (2.37)

where all factors V̄_{i} = V_{i}/V_{i+1} belong to Q, then V ∈ Q.

Proof.

(i) Let U ∈ Q and let V_{1} be a submodule; pick E ∈ UF(L), by hypothesis there is a C-H polynomial f(t) ∈ K[t] for π_{U}(E). Hence, v · f(π_{U}(E)) = 0 for all v ∈ U; in particular, V_{1} · f(π_{U}(E)) = 0, hence f(π_{V_{1}}(E)) = 0.

(ii) Similar to (i).

(iii) Let V_{1}, V_{2} ∈ Q and let U = V_{1} ⊗_{F} V_{2}, then for all a ∈ L

ad_{U}(a) = ad_{V_{1}}(a) ⊗ id_{V_{2}} + id_{V_{1}} ⊗ ad_{V_{2}}(a) .  (2.38)

Also, Pol(U) can be identified with Pol(V_{1}) ⊗_{F [L⃗]} Pol(V_{2}). Pick E ∈ UF(L), it belongs to an associative subalgebra generated by a finite number of generic elements x_{1}, …, x_{q} of L. Hence, by (2.38) the subalgebra of Pol(U) over K generated by E is is contained in A_{1}^{#} ⊗ A_{2}^{#}, where A_{i} is the subalgebra of Pol(V_{i}) over K generated by ad_{V_{i}}(x_{j}), j ∈ {1, …, q}. Thanks to lemma 2.8, A_{1}, A_{2} are finitely generated K-modules, hence so is A and (iii) holds.

(iv) Let U = V′ ⊕ V′′, where V′, V′′ ∈ Q; hence U has a chain

U = V′ ⊕ V′′ ⊇ V′ ⊇ 0

where all factors belong to Q; hence, (iv) will follow from (v).

(v) Pick E ∈ UF(L) and let f_{i}(t) be a C-H polynomial for πV̄_{i}(E). Then f(t) = f_{1}(t) … f_{m−1}(t) is a C-H polynomial for π_{U}(E). Indeed, for all i we have V_{i} · f_{i}(π_{V_{i}}(E)) ⊆ V_{i+1}, hence

V · f(π_{U}(E)) = V_{1} · f_{1}(π_{U}(E)) … f_{m−1}(π_{U}(E)) ⊆ V_{2} · f_{2}(π_{V_{2}}(E)) … f_{m−1}(π_{V_{2}}(E)) ⊆ …

⊆ V_{i} · f_{i}(π_{V_{i}}(E)) … f_{m−1}(π_{V_{i}}(E)) ⊆ … ⊆ V_{m} = 0 .

The lemma is proven. □

Now, let L be a finite dimensional k-sorted Lie algebra; put

V = V(L) = D + W ≃ ⊕_{i=1}^{k}D_{i} ⨁ W_{i} ,

so, it is an L-module; as a matter of fact, it is isomorphic to the quotient of L over the ideal ∑_{i=1}^{k}(M_{i} + C_{i}).

Proposition 2.2. Let L = (L; L_{1}, …, L_{k}) be a finite dimensional k-sorted Lie algebra, where Gn(L) generates L. Suppose that U is an L-module such that V ∈ Q(U), then Π(L) is integral over tr_{U}[L].

Proof. We need to show that the regular L-module belongs to Q(U); notice that L has a chain of submodules

L ⊇ N ⊇ N^{2} … ⊇ N^{m} = 0 ,

where m is the nilpotency index of N. By lemma 2.9 (v), we need only to prove that all factors lie in Q(U).

First of all, V_{i} = D_{i} + W_{i} is a submodule of V, hence, it belongs to Q(U). Next, L_{i}/N_{i} ≃ Gn_{i}(L) ∈ Q(U), because it is an extension of V_{i} by a trivial module S(L_{i})/N_{i} ≃ C_{i}. Next, N_{i} ⊆ N ∩ L_{i}, hence L_{i} + N/N ∈ Q(U) as quotient of L_{i}/N_{i}. Hence L/N = ∑_{i=1}^{k} L_{i} + N/N ∈ Q(U), because it is a quotient of the direct sum of L_{i} + N/N.

Next, N^{p}/N^{p+1} is a quotient of N̄^{⊗p}, where N̄ = N/N^{2}, so we need only to check that N̄ ∈ Q(U). By lemma 2.7 it is equal to W^{[0]}+⟨C⟩^{2}+N^{2}/N^{2}. Consider a chain of submodules

W_{1} ⊇ W_{2} ⊇ … ⊇ W_{n} ⊇ 0

where W_{j} = W^{[0]}+⟨C⟩^{j+1}+N^{2}/N^{2} and m−1 is the nilpotency index of C. Clearly, W_{j}/W_{j+1} is trivial module for all j < m, hence, we need only W_{m} = W^{[0]} +N^{2}/N^{2} ∈ Q(U). Consider a chain

U_{0} ⊇ U_{1} ⊇ … ⊇ U_{l} = 0

where U_{j} = W^{[j]} + N^{2}/N^{2}; by (2.29) for all j the quotient U_{j}/U_{j+1} is a sum of submodules being isomorphic to irreducible submodules in V, hence it lies in Q(U) which proves the proposition. □

6. Identities of irreducible representations

Take a variable y ∉ X and let L⟨y; X⟩ denote the subspace of all polynomials in L⟨X ∪ {y}⟩ that are linear with respect to y. Let V be a module over a Lie algebra L, we denote it by V : L; a polynomial f = f(y; x_{1}, …, x_{n}) ∈ L⟨y; X⟩ is called an identity of V : L, if f(v; a_{1}, …, a_{n}) = 0 for all v ∈ V and a_{1}, …, a_{n} ∈ L. Let T (V : L) be the subset of all identities of V : L; clearly, it is closed under substitution of Lie polynomials from L⟨X⟩ in place of variables from X and a polynomial from L⟨y; X⟩ in place of y. We call such a subspace of L⟨y; X⟩ a T-submodule. Obviously, every T-submodule can be identified with the set of all identities T (V : L) of an appropriate Lie module V : L.

In fact, it is equivalent to the language of Lie pairs, cf.[33]. Actually, let L̃ be the image of L in End(V) and let A denote the envelope of L̃; then T (V : L) consists of all polynomials of the form y · f(ad(x_{1}), …, ad(x_{n})), where f ranges over the ideal of identities of the Lie pair (A, L̃). Moreover, if I is weak T-ideal of A⟨X⟩, then

MI = {y · f(ad(x_{1}), …, ad(x_{n}))| f ∈ I}

is a T-submodule and every T-submodule has such a form; also put

LI = {x · f(ad(x_{1}), …, ad(x_{n}))| x ∈ X; f ∈ I} ,

it is a T-ideal of L⟨X⟩. Later we shall need Lie polynomials of the form

f(y; E_{1}, …, E_{n}, …) = y · g(E_{1}, …, E_{n}, …)

where g = g(x_{1}, …, x_{n}, …) ∈ A⟨X⟩ and E_{1}, …, E_{n} are polynomials in adjoint operators ad(x), x ∈ X.

The next fact is an analog of lemma 1.3.

Lemma 2.10. Let V be an irreducible finite dimensional module over a Lie algebra L and let f = f(y; x_{1}, …, x_{n}) be a multilinear nonidentity of V : L, f ∉ T (V : L).

Take a nonzero v ∈ V, then there exist a_{1}, …, a_{n} ∈ L and a polynomial E in adjoint operators ad_{V} (a),a ∈ L, such that v is equal to

f(v; a_{1}, …, a_{n})E  (2.39)

Proof. Notice first, that the subset u ∈ V such that f(u; L, …, L) = 0 is a submodule of V : L; as f ∉ T (V : L) and V is irreducible, it must be equal to 0. Hence, w = f(v; a_{1}, …, a_{n}) ≠ 0 for suitable a_{1}, …, a_{n} ∈ L and the submodule U generated by w coincides with V. Clearly, every element in U has form (2.39) for some E. □

In other words, the statement says that for every nonzero v ∈ V there is a polynomial of the form

g(y; x_{1}, …, x_{n}, x_{n+1}, …, x_{m}) = f · E  (2.40)

such that under a suitable substitution of x_{i} → x̄_{i} ∈ L and y → ȳ = v we get v = ḡ. Observe that g may be not linear with respect to x_{n+1}, …, x_{m}, but we can assume that it is a sum of multilinear polynomials.

Consider some examples.

Lemma 2.11. Let V be an irreducible L-module, dim(V) = l and dim(ad_{V} (L)) = m. Then MCap_{l^{2}} ⊈ T (V : L) and MCap_{m,1} ⊈ T (V : L).

Proof. Since V is irreducible and F is algebraically closed, the envelope A of ad_{V} (L) coincides with End(V) ≃ M_{l}(F) [35, p.152]. Recall (cf. lemma 1.3) that the verbal ideal Cap_{l^{2}}[A] is equal to A, in particular, V · Cap_{l^{2}}[A] ≠ 0, hence, MCap_{l^{2}} ⊈ T (V : L).

Next, pick

f = f(y; E_{1}, …, E_{l^{2}}, …) ∈ MCap_{l^{2}}

which is multilinear and skew-symmetric with respect to E_{1}, …, E_{l^{2}} and under a suitable substitution x → x̄ does not vanish. Observe that Ē_{1}, …, Ē_{l^{2}} must be a basis of A; we can replace them with elements of any another basis, hence, we can assume that the first m operators are adjoint ones, E_{i} = ad(x_{i}), where i ≤ m. □

So, every v ∈ V can represented as a specialization of polynomial (2.40), where f = f(y; E_{1}, …, E_{l^{2}}, …) is multilinear and skew-symmetric with respect to E_{1}, …, E_{l^{2}} or f = f(y; x_{1}, …, x_{m}, …) is multilinear and skew-symmetric with respect to x_{1}, …, x_{m}.

Now we go back to the canonical decomposition of a f.d. k-sorted Lie algebra L = (L; L_{1}, …, L_{k}). Recall that Gn_{i}(L) = D_{i} + C_{i} + W_{i} is a module over L; as N acts trivially, we may consider it as a module over L̄ = L/N. Note that L̄ is isomorphic to the direct sum of D ≃ D + N/N and C̄ = C + N/N. Hence, the regular L̄-module is the direct sum of simple summands D_{α}, α ∈ pr(D) = ⋃_{i=1}^{k} pr(D_{i}), and 1-dimensional trivial submodules of C̄.

Take β ∈ pr(W_{i}) ⊆ pr(W), the L̄-module W_{i,β} is said to be standard if it is isomorphic to D_{α} : L̄ for some α ∈ pr(D). In particular, it means that

[W_{i,β}, C̄] = 0 ,  (2.41)
[W_{i,β}, D_{γ}] = 0  (2.42)

for all γ ≠ α; besides, α ∈ pr(D_{i}) because [W_{i}, D_{j}] = 0 for j ≠ i.

For β ∈ pr(W_{i}) put ad_{β} = ad_{V}, where V = W_{i,β}. Suppose that W_{i,β} is non-standard, then, by Kushkulei-Rasmyslov’s theorem [23], T (W_{i,β} : L̄) = T (W_{i,β} : L) differs from T (D_{α} : L̄) = T (D_{α} : D_{α}) for all α ∈ pr(D).

Lemma 2.12. Suppose W = W_{i,β} is a non-standard submodule, where dim(ad_{β}(L)) = m. Then there exists ϕ = ϕ(y; x_{1}, …, x_{m}, …) ∉ T (W : L) which is multilinear and skewsymmetric with respect to y, x_{1}, …, x_{m}.

Proof. By lemma 2.11, there is a multilinear ψ_{1}(y; x_{1}, …, x_{m}, …) being skew-symmetric with respect to x_{1}, …, x_{m} such that

ψ̄_{1} = ψ_{1}(ȳ; x̄_{1}, …, x̄_{m}, …) ≠ 0

for some ȳ ∈ W and x̄_{j} ∈ L.

First, suppose that ad_{β}(L) is simple, hence, it is isomorphic to some D_{α}. Consider the case T (D_{α} : L̄) ⊈ T (W : L); by lemma 2.10, there exists

ψ_{2} = ψ_{2}(y; z_{1}, …, z_{n}) ∈ T (D_{α} : L̄)

which a sum of multilinear polynomials and ȳ = ψ_{2}(ȳ; z̄_{1}, …, z̄_{n}) for appropriate z̄_{1}, …, z̄_{n} ∈ L. Let ϕ denote the alternation of

ψ_{1}(ψ_{2}(y; z_{1}, …, z_{n}); x_{1}, …, x_{m}, …)

with respect to y, x_{1}, …, x_{m}, then under the same substitution ϕ̄ = m! · ψ̄_{1}. So, taking a suitable homogeneous component of ϕ, we get the required polynomial.

Thus, thanks to Kushkulei-Rasmyslov’s theorem, we can assume that

T (D_{α} : D_{α}) ⊉ T (W : L)

, hence, there exists

ψ_{2} = ψ_{2}(y; z_{1}, …, z_{n}) ∈ T (W : L)

such that ψ̄_{2} ≠ 0 under a suitable substitution x → x̄ of elements from D_{α} in place of y, z_{1}, …, z_{n}. Since dim(D_{α}) = dim(ad_{β}(L̄)) = m, we can find ψ_{1}(y; x_{1}, …, x_{m}, …) being multilinear and skew-symmetric with respect to x_{1}, …, x_{m} such that

ȳ = ψ_{1}(ȳ; x̄_{1}, …, x̄_{m}, …) .

Take v ∈ W such that [v, ψ̄_{2}] ≠ 0, hence

[v, ψ_{2}(ψ_{1}(ȳ; x̄_{1}, …, x̄_{m}, …); z̄_{1}, …, z̄_{n})] ≠ 0 ;

if we swap v and any x̄_{i}, we shall get 0 by the choice of ψ_{2}. Hence, the alternation of the polynomial

[y, ψ_{2}(ψ_{1}(x; x_{1}, …, x_{m}, …); z_{1}, …, z_{n})]

with respect to y, x_{1}, …, x_{m} is not an identity of W : L; choosing a suitable homogeneous component, we get the required polynomial.

Now, suppose that ad_{β}(L) is not simple, then it has nonzero ideals L′, L′′ (maybe equal) such that [L′, L′′] = 0. We can choose z̄′_{j} ∈ L′, z̄′′_{j} ∈ L′′, j + 1, …, t and ỹ ∈ W such that ȳ = ∑_{j}[[ỹ, z̄′_{j}], z̄′′_{j}]. Then the alternation ϕ of

ψ_{1}([[y, z′_{1}], z′′_{1}]; x_{1}, …, x_{m}, …)

with respect to y, x_{1}, …, x_{m} is not an identity of W : L. □

7. ”Counter-example”

Like in the associative case we need the notion of “counter-example”. Let A = (L; V) be a finite dimensional g.Lie module from 𝔐_{k,s,r} such that there exists a strictly ascending chain of T-ideals in Q(A) starting with T (A). Then one can find a sequence of multlinear polynomials f_{1}, f_{2}, …, f_{n}, … ∈ Q(A), deg(f_{i}) < deg(f_{i+1}), such that the chain of Tideals:

T (A) ⊂ Γ_{1} ⊂ Γ_{2} ⊂ … ⊂ Γ_{n} ⊂ …  (2.43)

where Γ_{n} = (f_{1}, f_{2}, … f_{n})_{⊤} + T (A), ascends strictly.

Let Γ_{n}^{+} denote the T-ideal generated by T (A) and all homogeneous polynomials from Γ_{n} of degree greater than deg(f_{n}). Put Γ = ∑_{n=1}^{∞} Γ_{n}^{+} + T (A), then for all n the identity f_{n} does not belong to Γ, but every ”proper” consequence of f_{n}, i.e., a homogeneous polynomial from (f_{n})_{⊤} of degree > deg(f_{n}), lies in Γ. We call such a sequence appropriate.

A T -ideal Γ is called critical if it has an appropriate sequence; the T-ideal generated by Γ and the appropriate sequence is denoted by Γ^{#}. Like in associative algebras, the main property of a critical T-ideal is that it can not be the ideal of identities of a finite dimensional g.Lie module.

Lemma 2.13. If Γ is a critical T-ideal, then Γ ≠ T (A), where A is a finite dimensional g.Lie module.

Proof. Suppose that Γ = T (A), where A = (L, V) is a finite dimensional g.Lie module. Let f_{1}, …, f_{n}, … be the appropriate sequence for Γ. For all n we have f_{n}(x_{1}, x_{2}, …, x_{m}) ∉ T (A), hence one can find x̄_{1}, x̄_{2}, …, x̄_{m} ∈ L, such that f̄_{n} = f_{n}(x̄_{1}, …, x̄_{m}) ≠ 0.

By lemma 2.4 we have

C + [L, L] = ∑_{i=1}^{k} C_{i} + [L_{i}, L] = ∑_{i=1}^{k} L_{i} = L ;

since f_{n} is multilinear, we may assume that x̄_{j} ∈ C_{j} ∪ [L_{i}, L] provided x_{j} ∈ X_{i}. Observe that x̄_{j} cannot be in [L_{i}, L]; actually, the substitution a commutator [a, b] ∈ L_{i}⟨X⟩ in place of x_{j} means that f̄_{n} is equal to the value of a “proper” consequence of f_{n} which belongs to T (A), therefore f̄_{n} must be equal to 0. So, for all n we can replace the variables of f_{n} by suitable elements from ⋃_{i=1}^{k} C_{i} and get a nonzero element f̄_{n} ∈ V. Let c_{1}, …, c_{t} be the union of bases of C_{i}, i ∈ {1, …, k}; since f_{n} are multilinear we may assume that these elements are among c_{1}, …, c_{t}.

Consider a set of free variables Y = {y_{1}, y_{2}, …, y_{t}}, Y_{j} = {y_{i} ∈ Y | c_{i} ∈ C_{j}} and the free g.Lie module

M⟨Y ⟩ = (L⟨Y ⟩; V ⟨Y ⟩) ,

and define a homomorphism (ϕ, ψ) from M⟨Y ⟩ to A which takes y_{i} to c_{i}; then the image coincides with the g.Lie submodule B generated by c_{1}, …, c_{t}.

Furthermore, put I = L⟨Y ⟩^{q}, where ⟨C⟩^{q} = 0 (recall that ⟨C⟩ is nilpotent), then Ker(ϕ) contains I; besides, let J denote the ideal of identities of B, then J ⊆ Ker(ψ). Hence, (ϕ, ψ) induces an epimorphism (ϕ̃, ψ̃) from

M̃⟨Y ⟩ = (L̃; V ⟨Y ⟩/J)

where L̃ = L⟨Y ⟩/I, onto B. We claim that Ṽ = V ⟨Y ⟩/J satisfies a.c.c. for submodules.

Lemma 2.14. Let A = (L; V) be a g.Lie module over a commutative noetherian ring K. Suppose that L is a finitely generated module over K, then En(A) is a noetherian algebra (a.c.c. for right ideals) and, therefore, V is a noetherian module over En(A).

Proof. Recall that ν(L) = ∑_{i=1}^{k} ν_{i}(L) is a Lie subalgebra of End_{Φ}(V)^{(−)} and En(A) is its envelope, hence, lemma follows from [11, sec.2.3.8]). □

Observe that L̃ is finite dimensional over F, hence the lemma gives the necessary property for Ṽ.

Recall that f_{n} does not vanish under replacing all variables x_{i} with the corresponding basis elements c_{i_{j}}. Let g_{n} ∈ V ⟨Y ⟩ be the polynomial f_{n} where x_{i} is replaced with y_{i_{j}}. Then it is homogeneous of the same degree as f_{n} and obviously ψ(g_{n}) is equal to f̄_{n} ≠ 0.

Now let Ṽ_{n} be the submodule of Ṽ over En(M̃⟨Y ⟩) generated by g̃_{1}, …, g̃_{n}, where g̃_{n} = ψ̃(g_{n}) = g_{n}/J. Since Ṽ satisfies a.c.c. for submodules, Ṽ_{n} = Ṽ_{n+1} for some n, hence g̃_{n+1} ∈ Ṽ_{n} and

g_{n+1} − g_{n}E ∈ J

for some E ∈ En(M̃⟨Y ⟩). Moreover, since g_{n+1}, g_{n} are homogeneous elements and J is a homogeneous submodule, we may assume that E is homogeneous of degree deg(g_{n+1}) − deg(g_{n}) > 0. Observe that g_{n}E is obtained from f_{n}E by the corresponding substitution of variables from Y (since X is infinite, we may assume that Y ⊆ X and no variables from Y occurring in any f_{n}). By the definition of the appropriate sequence, f_{n}E ∈ Γ = T (A), hence, ψ(g_{n}E) = 0 and ψ(g_{n+1}) must be equal to 0, which is impossible. Thus, the lemma is proven. □

The next statement is an analog of lemma 1.6 and one can prove it in the same way.

Proposition 2.3. Let Γ, Γ_{1} be T-ideals, where Γ is critical. Then Γ + Γ_{1} is critical, where the appropriate sequence is a subsequence of that of Γ, or Γ∩Γ_{1} is a critical T-ideal with an appropriate sequence in Γ_{1}.

Now we are prepared to define the main object; a pair of T-ideals T (A) ⊂ Γ is called a ”counter-example” iff

(i) A is a finite dimensional generalized Lie module satisfying (2.20); (ii) Γ is a critical T-ideal, where Γ^{#} ⊆ Q(A). The next statement is an analog of lemma 1.7.

Lemma 2.15. Let Γ be a critical T -ideal and let B_{j}, j = 1, …, t be finite dimensional g. Lie modules such that Γ^{#} ⊆ Q(B_{j}) for all j and

Γ ∩ ⋂_{j=1}^{t} T (B_{j})

has no an appropriate sequence in Γ^{#} ∩ ⋂_{j=1}^{t} T (B_{j}). Then there is a T-ideal Γ_{1} with the appropriate sequence in Γ^{#} such that T (B_{j}) ⊂ Γ_{1} ⊆ Γ + T (B_{j}) for some j.

Proof. Put Υ_{0} = Γ and Υ_{m} = Γ ∩ ⋂_{j=1}^{m} T (B_{j}), Υ_{m}^{#} = Γ^{#} ∩ ⋂_{j=1}^{m} T (B_{j}) for m ≥ 1. Take the maximal m such that Υ_{m} has an appropriate sequence in Υ_{m}^{#}; then 0 ≤ m < t. By the proposition above Γ_{1} = Υ_{m} + T (B_{m+1}) is a critical with the appropriate sequence in Υ_{m}. □

8. Type of a ”counter-example”

In the associative case the type of a ”counter-example” T (A) ⊂ Γ is defined as a pair of integers ≥ 0 characterizing the difference between T (A) and Γ, and the lexicographical order on ℕ_{0} × ℕ_{0} enables us to carry out the induction on the type. In this section we define the type

type(A, Γ)

where A = (L; V) is a finite dimensional g.Lie module from M_{k,s,r} satisfying (2.20), with a fixed canonical decomposition of L, and Γ is a T-ideal in Q(A) such that Γ ⊈ T (A).

Clearly, the definition of type should be based on certain numerical characteristics of A; in fact, we will use simple components of the L-modules Gn_{i}(L). Since the dimensions of irreducible modules over a finite dimensional Lie algebra are not bounded, finite sequences of an arbitrary length should be employed instead of pairs (or n-tuples); we will use an inductive construction of well-ordered sets with the lexicographical order.

Let {M_{i}}_{i∈ℕ} be a sequence of well-ordered sets, where 0_{i} denotes the least element of M_{i}. Consider the set Fin(M_{i}; i ≥ 1) of finite sequences a = (a_{1}, …, a_{i}, …) (i.e., there exists i_{0} such that for all i > i_{0} we have a_{i} = 0_{i}). Define a linear order on Fin(M_{i}; i ∈ ℕ): for all a, b put a > b if there is k such that a_{k} > b_{k} and for all n > k we have a_{n} = b_{n}. Clearly, it is the direct limit of the lexicographically ordered sets M_{1} × … × M_{q}, q ∈ ℕ; moreover, every ascending chain of elements in Fin(M_{i}; i ∈ ℕ) is contained in such subset, hence we get immediately the following statement.

Lemma 2.16. Fin(M_{i}; i ∈ ℕ) is a well-ordered set.

In fact, we shall mostly deal with powers M^{∞} of a well-ordered set M, i.e., M^{∞} = Fin(M_{i}; i ∈ ℕ), where M_{i} = M for all i. In particular , a finite power M^{k} is a well-ordered subset of M^{∞} (note that this order is lexicographical from the right to the left).

Like in the associative case, type(Γ, A) should measure the difference Γ\T (A). Recall that over a field of characteristic 0 any T-ideal is generated by its multilinear polynomials, hence, we need to estimate the size of the multilinear part of Γ\T (A). If f ∈ Γ is a multilinear nonidentity of A, then f does not vanish under some substitution of elements from A in place of variables. In fact, we do not need to consider all such substitutions.

Lemma 2.17. Let f ∈ Q(A) be a nonidentity of A, then there exists a multilinear polynomial g = g(x_{1}, …, x_{n}) ∈ (f)_{⊤} such that g does not vanish under a substitution x_{i} → x̄_{i}, where x̄_{i} ∈ Gn_{j}(L), if x_{i} ∈ X_{j}.

Proof. Suppose f̃ = f(ỹ_{1}, …, ỹ_{m}) ≠ 0 for suitable elements ỹ_{i} ∈ L_{j}. By lemma 2.6 modulo I_{j} the Lie algebra L_{j} is generated by Gn(L_{j}), hence ỹ_{i} has the form of a Lie polynomial h_{i} ∈ L⟨X_{j}⟩ ⊆ L_{j}⟨X⟩ in some elements of Gn(L_{j}). Hence, f_{1} = f(h_{1}, …, h_{m}) does not vanish under substitution of the same elements. Taking the linearization of a suitable homogeneous part of f_{1}, we get the required polynomial g. □

So, we should consider “nonzeros” of multilinear polynomials from Γ consisting of elements in ⋃_{i=1}^{k} Gn(L_{i}). Moreover, thanks to linearity such elements may be taken only from simple components. Put

Spl(L_{i}) = ( ⋃_{α∈pr(D_{i})} D_{i,α}) ∪ ( ⋃_{β∈pr(W_{i})} W_{i,β}) ∪ C_{i} ,

Spl(L) = ⋃_{i=1}^{k} Spl(L_{i}) .

Let f(x) = f(x_{1}, …, x_{n}) ∈ V ⟨X⟩, where x means the n-tuple of variables (x_{1}, …, x_{n}). A substitution x → x̄, where x̄ ∈ L_{i} provided x ∈ X_{i}, defines a homomorphism of the free module to A; the image of f is f̄ = f(x̄) = f(x̄_{1}, …, x̄_{n}). We will consider mostly substitutions x → x̄ such that

x̄ ∈ Spl(L)  (2.44)

for all x ∈ X. Put

V(f, A) = {x̄| f(x̄) ≠ 0} ,  (2.45)

where x → x̄ satisfies (2.44) and given a T -ideal Γ of M⟨X⟩ let

V(Γ, A) = ⋃_{f} V(f, A) ,  (2.46)

where f runs over all multilinear polynomials in Γ.

Take x̄ = (x̄_{1}, …, x̄_{n}) ∈ V(f, A) and let π^{(m)}(x̄) be the set of all projections α ∈ pr(D) such that

dim(D_{α}) = m and ∃j [x̄_{j}, D_{α}] ≠ 0 .

Put

a^{(m)}(x̄) = |π^{(m)}(x̄)|  (2.47)

(so, a^{(m)}(x̄) is the total number of such D_{α}, not variables x_{j} with x̄_{j} ∈ D_{α}!),

a(x̄) = (a^{(1)}(x̄), a^{(2)}(x̄), …, a^{(m)}(x̄), …).  (2.48)

As dim(D) < ∞, it is a finite sequence, therefore, it belongs to the well-ordered set ℕ_{0}^{∞}. Of course, for some natural m simple Lie algebras of dimension m do not exist and, therefore, always a^{(m)} = 0.

At the next stage we count the number of variables x_{j} with x̄_{j} ∈ N, the number of these is bounded by the nilpotency index of the radical of En(A). Actually, En(A) is the envelope of ν(L) and by (2.10)-(2.12) ν(N) lies in the solvable radical of the commutator subalgebra of ν(L), which is contained in the radical of En(A).

Next, let b^{(m,0)}(x̄) denote the number of x̄_{j} ∈ W_{i,β}, where W_{i,β} is a standard L̄- module with

dim(ad_{β}(L̄)) = m .  (2.49)

Also, let b^{(m,l)}(x̄) be the number of x̄_{j} ∈ W_{i,β} for a nonstandard Lie submodule W_{i,β} of dimension l satisfying (2.49). Put

b^{(m)}(x̄) = (b^{(m,0)}(x̄), b^{(m,1)}(x̄), …, b^{(m,l)}(x̄), …) ∈ ℕ_{0}^{∞};  (2.50)
b(x̄) = (b^{(1)}(x̄), b^{(2)}(x̄), …, b^{(m)}(x̄), …),  (2.51)

it is a element of the well-ordered set (ℕ_{0}^{∞})^{∞}.

For a ∈ ℕ_{0}^{∞},

a = (a^{(1)}, a^{(2)}, …, a^{(m)}, …) ,  (2.52)

put

∥a∥ = ∑_{m=1}^{∞} m · a^{(m)} ;  (2.53)

also, if

b = (b^{(1)}, b^{(2)}, …, b^{(m)}, …) ∈ (ℕ_{0}^{∞})^{∞},  (2.54)

where

b^{(m)} = (b^{(m,0)}, b^{(m,1)}, …, b^{(m,l)}, …) ,

then put

|b| = ∑_{m=1}^{∞} ∑_{l=0}^{∞} b^{(m,l)} ,  (2.55)

Thus, |b(x̄)| is the total number of occurences of elements from W (see lemma 2.5), and ∥a(x̄)∥ is equal to the total dimension of simple components D_{i,α} with [x̄_{j}, D_{i,α}] ≠ 0 for some x_{j}.

Since ∥a(x̄)∥ is not greater than the dimension of the semisimple part D, the set {a(x̄)| x̄ ∈ V(A, Γ)} is finite; consider the subset of such elements with maximal ∥a(x̄)∥ and let a(A, Γ) denote the maximal element. By definition, we have

∥a(x̄)∥ ≤ ∥a(A, Γ)∥ ≤ dim_{F} (D) .  (2.56)

Similarly, |b(x̄)| does not exceed the nilpotency index of the radical of En(A). Hence, the set {b(x̄)| x̄ ∈ V(Γ, A)} is finite; we denote by b(Γ, A) the greatest element.

Now, let a, b be elements of form (2.52), (2.54) respectively; also, let c ∈ ℕ. We are going to define a T-ideal

S_{a,b,c}(Γ) ⊆ Γ .  (2.57)

In fact, it will be an operator on the set of T-ideals; we will write down it in terms of operators in E(X), see (2.16).

First of all, if ∥a∥ = |b∥ = 0, then we put (2.57) equal Γ. Next, suppose ∥a∥ ≠ 0 or |b| ≠ 0.

Step 1. Take a multilinear polynomial f ∈ Γ with a set ϑ ⊆ varb(f). Formally, in terms of E(X), it means that f ∈ τ · Γ, where τ is the projector on a homogeneous component of V ⟨X⟩ consistsing of multilinear polynomials of degree 1 with respect to every x ∈ ϑ.

Step 2. For all x ∈ ϑ take multilinear f_{x} ∈ L_{i}⟨X⟩ such that varb(f_{x}) ∩ varb(f_{y}) = ∅, where x ≠ y, and ⋃_{x∈ϑ} varb(f_{x}) contains a family of disjoint subsets Θ_{t}, t ∈ {0, …, |b|}, where

|Θ_{0}| = ∥a∥,  (2.58)
|Θ_{t}| = ∥a∥ + 1  (2.59)

for all t ≥ 1 .

Next, for all x ∈ ϑ take multilinear

g_{x} ∈ ∑_{p≥c}[f_{x}, \underbrace{L_{i}⟨X⟩, …, L_{i}⟨X⟩}_{p}]  (2.60)

with disjoint sets of variables varb(g_{x}), such that the polynomial

f_{1} = f|_{∀x=g_{x}}

is multilinear.

Since ∥a∥ ≠ 0 or |b| ≠ 0, ϑ must be nonempty, because Θ_{0} ≠ ∅ or Θ_{1} ≠ ∅.

Alternate f with respect to variables of Θ_{t} for each t, and let h_{0} denote the resulting polynomial. More precisely, let G be the product of groups (in the group Sym(X)),

G = ∏_{t=0}^{|b|} Sym(Θ_{t}) ,  (2.61)

it is the direct product, because the corresponding subsets are disjoint. It acts on f_{1} by permutation of variables; observe that σ · f_{1} ∈ Γ for all σ ∈ G thanks to the specific form of every g_{x}: since c ≥ 1, g_{x} depends on some y ∈ X_{i} (where x ∈ X_{i}) which is fixed under the action of G. So, h_{0} = S_{G} · f_{1}, where

S_{G} = ∑_{σ∈G} sgn(σ)σ ,  (2.62)

although S_{G} ∉ E(X), its composition with the substitution x → g_{x} and the projector τ from Step 1 does belong to E(x).

Step 3.

The element a ∈ ℕ_{0}^{∞} defines a partition of Θ_{0}: we choose a sequence of disjoint subsets, where the m-th term contains m · a^{(m)} elements. Observe that the total number of elements is ∥a∥ = ∑_{m=1}^{∞} m · a^{(m)}, therefore it is a partition of Θ_{0} indeed. Next, every variable x from the m-th subset is replaced with a multilinear Lie polynomial

ϕ_{x} = ψ′_{x} · ψ_{x},  (2.63)

where ψ_{x} and ψ′_{x} are skew-symmetric with respect to m variables, and the total degree of ϕ_{x} is ≥ c; clearly, we suppose that ϕ_{x} ∈ L_{i}⟨X⟩ provided x ∈ X_{i}.

Now we make a similar substitution for elements in Θ_{t}, t ≥ 1. First, we fix a variable y_{p} ∈ Θ_{t}; then the order of Θ_{t}\{y_{p}} is equal to ∥a∥, and we make a substitution of the same form as in the case of Θ_{0}. So, we need only to define the substitution for y_{p}.

The element b defines a partition of the set {1, …, |b|}: we fix disjoint subsets η^{(m,l)} such that |η^{(m,l)}| = b_{i}^{(m,l)} for all m, l. For every p ∈ η^{(m,0)} we replace the fixed variable y_{p} ∈ Λ_{i,p} with the polynomial ϕ_{y_{p}} of form (2.63). If p ∈ η^{(m,l)}, where l ≥ 1, then y_{p} is replaced with a polynomial of the form

ϕ_{y_{p}} = ψ′_{y_{p}}(ψ_{y_{p}}; E_{1}, … E_{l^{2}}, …) ,  (2.64)

where ψ_{y_{p}} is skew-symmetric with respect to m + 1 variables, and ψ′_{y_{p}} is skew-symmetric with respect to E_{1}, …, E_{l^{2}}. Also, the degree of ϕ_{x} is ≥ c and ϕ_{y_{p}} ∈ L_{i}⟨X⟩ if y_{p} ∈ X_{i}.

In fact, the substitution x → ϕ_{x} induces an endomorphism of the free g.Lie module M⟨X⟩; denote by ϱ_{a,b,c} the corresponding endomorphism of V ⟨X⟩, so it belongs to E(X) and

h = ϱ_{a,b,c}(h_{0}) ;  (2.65)

besides, we take ϕ_{x} depending on different variables, so h is multilinear.

Finally, let S_{a,b,c} ∈ E(X) denote the composition of τ ∈ E(X) from Step 1, x → g_{x} , S_{G} and ϱ_{a,b,c}, then

h = S_{a,b,c}(f) .  (2.66)

Note that for fixed a, b, c we have a class of operators of such form; define (2.57) as the T-ideal generated by (2.66), where S_{a,b,c} ranges over this class. Since Γ is a T-ideal, (2.57) is contained in Γ. Also, an element of (2.66) is called a typical generator of (2.57); recall that it is multilinear.

Lemma 2.18. Let a = a(Γ, A) and let S denote the set of all b ∈ (ℕ_{0}^{∞})^{∞} such that for all c ∈ ℕ the T-ideal (2.57) is not contained in T (A). Then S is finite.

Proof. Take all c ≥ 2; there is a typical generator (2.66) not vanishing under some substitution x → x̄; we can assume that x̄ ∈ V(h, A). Indeed, like in lemma 2.17, we can replace each x̄ with a Lie polynomial h_{x} in elements from Spl(L). We can assume that h_{x} is the sum of multilinear ones depending on different variables, hence every homogeneous component of

h′ = h|_{x=h_{x} ∀x}

is multilinear and obtained from h by replacing every x with a homogeneous component of h_{x}. From the definition of (2.66) it follows that all homogeneous components of h′ are typical generators of (2.57) and one of these does not vanish under some x̄ ∈ V(Γ, A).

We use notation given above; consider a subset Θ_{p}, p ≥ 1; under x → x̄ for all y ∈ Θ_{p} the polynomial ϕ_{y} takes value in D ∪ N. Indeed, if there is x ∈ varb(ϕ_{y}) such that x̄ ∈ W, then ϕ̄_{y} ∈ N; otherwise x̄ ∈ D for all x ∈ varb(ϕ_{y}), because D commutes with C and the degree of ϕ_{y} is greater or equal to the nilpotency index of ⟨C⟩.

Collect all x ∈ Θ_{p} such that ϕ̄_{x} ∈ D, then, by definition of a(x̄), they all belong to a subalgebra of D of dimension ≤ ∥a(x̄)∥ ≤ ∥a(Γ, A)∥ = ∥a∥ thanks to (2.56 ). Since Θ_{p} has ∥a∥ + 1 elements, there is y ∈ Θ_{p} such that ϕ̄_{y} ∈ N. Hence, |b|, the total number of the subsets Θ_{p}, cannot exceed the nilpotency index of the radical of En(A). Therefore, all coordinates of b are bounded. □.

Put

t(Γ, A) = max(S) ,  (2.67)

where S is the set defined in the lemma above; if S is empty, then we let t(Γ, A) be equal to the least element of (ℕ_{0}^{∞})^{∞}. From the definition we get immediately the following.

Lemma 2.19. Let B be a finite dimensional g.Lie module with a fixed canonical decomposition such that Γ ⊈ T (B). Suppose that T (A) ⊆ T (B) and a(Γ, B) = a(Γ, A), then

t(Γ, B) ≤ t(Γ, A) .

Next, take β ∈ pr(W), then β ∈ pr(W_{i}) for some i; by definition, W_{i,β} is contained in the weight subspace N_{i}(λ), hence, there exists l ∈ ℕ such that W_{i,β}^{[l]} = 0; such a minimal l is called the height of W_{i,β} and denoted by ht(W_{i,β}). Also, if x̄_{j} ∈ W, then x̄_{j} ∈ W_{i,β} (we consider x̄ ∈ V(Γ, A)); put ht(x̄_{j}) = ht(W_{i,β}). Then ht(x̄) is defined as the the sum of all ht(x̄_{j}), j ∈ {1, …, n}, with x̄_{j} ∈ W. Since ht(x̄) is not greater than |b(x̄)| · m, where m is the maximum of heights of W_{i,β}, β ∈ pr(W), we can put

ht(Γ, A) = max{ht(x̄)| x̄ ∈ V(Γ, A)} .  (2.68)

Finally, put

type(Γ, A) =  (2.69)
= (ht(Γ, A) , b(Γ, A) , |b(Γ, A)| , t(Γ, A) , a(Γ, A) , ∥a(Γ, A)∥) ,

it is an element of the following well-ordered set

ℕ_{0} × (ℕ_{0}^{∞})^{∞} × ℕ_{0} × (ℕ_{0}^{∞})^{∞} × ℕ_{0}^{∞} × ℕ_{0} .  (2.70)

By definition, we have

type(Γ, A) = type(Γ + T (A), A) ;  (2.71)

besides, let Γ_{1} ⊆ Q(A) be a T-ideal containing Γ, then it follows from the definition that

type(Γ, A) ≤ type(Γ_{1}, A) .  (2.72)

So, the type of a ”counter-example” T (A) ⊂ Γ is defined as type(Γ, A); since the definition depends on the canonical decomposition, we can choose one with respect to which the type takes the smallest value.

CHAPTER 3 Finite basis of identities in finite dimensional Lie algebras

In this chapter, using the induction on the type of ”counter-example”, we will prove theorem 2.2 and thereby the existence of a finite basis of identities of PI representations of Lie algebras over a field of characteristic zero.

1. Linearization of a “counter-example”

Recall that type(Γ, A) is an element of the well-ordered set (2.69); hence, we can compare ”counter-examples” from different classes of g.Lie modules (even over different algebraically closed fields). Let T (A) ⊂ Γ be a ”counter-example” of the smallest type in some variety of g.Lie modules 𝔐_{k,s,r} over an algebraically closed field F and our goal is to construct a “counter-example” of a smaller type in a variety 𝔐_{k′,s′,r′} over an algebraically closed field extension of F.

In the first two sections we will replace the given “counter-example” with another one of some canonical form.

Recall that b(A, Γ) is the maximum of b(x̄), where x̄ runs over V(A, Γ); to find b(x̄) we count x̄_{j} in irreducible components of W_{i}, and it would be convenient to have them in different ones; this is the motivation of the following definition.

A finite dimensional g. Lie module is said to be linearized, if for all β ∈ pr(W) the square of the ideal of En(A) generated by ν(W_{β}) is equal to 0.

Take x̄ ∈ V(Γ, A) and put

π(x̄) = ⋃_{m≥1} π^{(m)}(x̄) ,  (3.1)

it is the set of all projectors α ∈ pr(D) such that [D_{α}, x̄] ≠ 0 for some x ∈ varb(f). Also, let π_{i}(x̄) = π(x̄) ∩ pr(D_{i}). Next, let ω_{i}(x̄) be the collection of all projectors β ∈ pr(W_{i}) such that x̄ ∈ W_{i,β} for some x ∈ varb(f). Also, for each β ∈ ω_{i}(x̄) put m_{β} = m_{β}(x̄) to be the number of such variables in varb_{i}(f); if A is linearized, then m_{β} = 1 for all β. Finally, set ω(x̄) = ⋃_{i=1}^{k} ω_{i}(x̄).

Proposition 3.1. There exists a ”counter-example” T (A) ⊂ Γ of the minimal type such that

(1) A is linearized;

(2) there exists x̄ ∈ V(Γ, A) such that

π(x̄) = pr(D) , ω(x̄) = pr(W) .  (3.2)

Proof. Take x̄ ∈ V(Γ, A) and put

ξ_{i}(x̄) = π_{i}(x̄) ∪ {(β, m_{β})| β ∈ ω_{i}(x̄)} ,

where (β, m_{β}) means a pair of elements; then ξ(x̄) denotes the k-tuple (ξ_{1}(x̄), …, ξ_{k}(x̄)). Obviously, the set

J = {ξ| ∃x̄ ∈ V(Γ, A) , ξ = ξ(x̄)}

is finite.

Lemma 3.1. For each ξ ∈ J there exists a linearized finite dimensional g.Lie module

A^{ξ} = (L^{ξ}; V^{ξ})

such that

T (A) = Γ ∩ ⋂_{ξ∈J} T (A^{ξ})  (3.3)

and for all x̄ ∈ V(Γ, A) if ξ = ξ(x̄), then for all i

D_{i}(L^{ξ}) ≃ ⨁_{α∈π_{i}(x̄)} D_{i,α} ,  (3.4)

W_{i}(L^{ξ}) ≃ ⨁_{β∈ω_{i}(x̄)} m_{β}W_{i,β} .  (3.5)

Proof. Let P_{i} denote the sum of D_{i,α}, where α ∈ π_{i}(x̄); also, put P = ∑_{i=1}^{k} P_{i} ⊆ D. Take β ∈ ω_{i}(x̄) and for every j ∈ {1, …, m_{β}} denote by U_{i,β,j} the j-th copy of W_{i,β}, so

U_{i,β} = \underbrace{W_{i,β} ⨁ … ⨁ W_{i,β}}_{m_{β}}

is equal to the direct sum of U_{i,β,j}, j = 1, …, m_{β}. Finally, let

U_{i} = ⨁_{β∈ω_{i}(x̄)} U_{i,β}

and U = ⨁_{i=1}^{k} U_{i}.

Consider the k-sorted subalgebra Q = (Q; Q_{1}, …, Q_{k}) generated by ⋃_{i=1}^{k}(P_{i} + C_{i}), then Q = P + ⟨C⟩ and Q_{i} is the sum of P_{i} and the ideal of ⟨C⟩ generated by C_{i}. The Q-submodule generated by W_{i,β} is equal to W_{i,β}^{[0]}; denote by Ũ_{i,β,j} the j-th copy of this module, it is generated by U_{i,β,j} ⊆ Ũ_{i,β,j}. Furthermore, fix a basis Z_{i,β,j} of U_{i,β,j}; then Ũ_{i,β,j} has a basis Z̃_{i,β,j} equal to a disjoint union of Z_{i,β,j} and subsets of the form

Z_{i,β,j}(ad(c_{1}) − λ(c_{1})) … ad(c_{l}) − λ(c_{l})) ⊆ Ũ_{i,β}^{{l}} .  (3.6)

Put

Ũ_{i,β} = ⨁_{j=1}^{m_{β}} Ũ_{i,β,j} , Ũ_{i} = ⨁_{β∈ω_{i}(x̄)} Ũ_{i,β} , Ũ = ⨁_{i=1}^{k} Ũ_{i} ;

then

Z̃_{i} = ⋃_{β∈ω_{i}(x̄)} ⋃_{j=1}^{m_{β}} Z̃_{i,β,j}

is a basis of Ũ_{i} and Z̃ = ⋃_{i=1}^{k} Z̃_{i} is that of Ũ.

Consider the free k-sorted Lie algebra L⟨Z̃⟩ with Z̃ as the set of free generators; then Ũ = vect(Z̃) ⊆ L⟨Z̃⟩. We can extend the action of Q from Ũ to L⟨Z̃⟩, where every element of Q defines a derivation. Hence, the direct sum of vector spaces M = Q ⊕ L⟨Z̃⟩ is equipped with a Lie algebra structure, where Q is a subalgebra and L⟨Z̃⟩ is a Q-module (therefore, L⟨Z̃⟩ is an ideal of M). Also note that U and Q generate M as a Lie algebra.

Let M(Z_{i,β,j}, t) denote the subspace of M of all elements homogeneous with respect to Z̃_{i,β,j} of total degree t. In other words, f ∈ M(Z_{i,β,j}, t) iff for all γ ∈ F we have f = γ^{t}f under replacing every basis element z ∈ Z̃_{i,β,j} with γz. In fact, this mapping is an endomorphism of the Lie algebra M; actually, the simulteneous scaling of all basis vectors from Z_{i,β,j} induces an endomorphism of the free Lie algebra L⟨Z̃⟩ which agrees with the Q-module structure (i.e., it is a endomorphism over Q). Hence, every homogeneous component is a Q-submodule.

Next, let q ∈ ℕ, q ≥ 2 such that N^{q} = 0, then L⟨Z̃⟩^{q} is an ideal of M; put

M̄ = M/L⟨Z̃⟩^{q} .

Notice that M̄ inherits the homogeneity with respect to each Z_{i,β,j}. Besides, q ≥ 2, hence this canonical homomorphism is monic on Q and Ũ, so we can identify them with their images in M̄. Let M̄_{i} denote the ideal generated by Q_{i} and Ũ_{i}, then M̄ = (M̄; M̄_{1}, …, M̄_{k}) is a k-sorted Lie algebra.

By definition, P ⊆ Q is a maximal semisimple subalgebra of M̄ and the solvable radical S(M̄) is generated by Ũ and C; in particular, M̄ is finite dimensional. Next, by lemma 2.5, W_{i,β} ≃ U_{i,β,j} is a nontrivial L-module; recall that P includes all simple parts acting nontrivially on W_{i,β}, hence Ũ is contained in the commutator subalgebra [M̄, M̄]. Hence, N(M̄) = S([M̄, M̄]) is generated as an algebra by U and ⟨C⟩^{p}, p ≥ 2; in particular, M̄ satisfies (2.20).

Observe that P_{i} is a maximal semisimple subalgebra of M_{i} and the solvable radical of M_{i} is equal to L_{i}⟨Z̃⟩/L⟨Z̃⟩^{q}; therefore, we can put

D_{i}(M̄) = P_{i} .

Let N′_{i} denote the sum of ⟨C_{i}⟩^{2}, I_{i}(M̄) = ∑_{j≠i}[M̄_{i}, M̄_{j}] and the subalgebra generated by Ũ_{i}. By lemma 2.5, N_{i}(M̄) contains Ũ_{i}, hence, N′_{i} ⊆ N_{i}(M̄) and therefore, S(M_{i}) = N_{i}(M̄) + C_{i} = N′_{i} + C_{i}. On the other hand, N_{i}(M̄) ∩ C_{i} = (C_{i}, C) ∩ C_{i} = 0, hence, N_{i}(M̄) = N′_{i} and C_{i} completes N_{i}(M̄) to S(M̄_{i}); thus, we can put

C_{i}(M̄) = C_{i} .

Finally, U_{i} completes

M_{i}(M̄) = N_{i}(M̄)^{[1]} + ⟨C_{i}⟩^{2} + N_{i}(M̄)^{2} + I_{i}(M̄)

to N_{i}(M̄), and we put

W_{i}(M̄) = U_{i} .

Thus,

Gn_{i}(M̄) = P_{i} + C_{i} + U_{i} .  (3.7)

Now, let B denote the universal envelope U(M̄, s); since every endomorphism of M̄ induces that of B, the algebra B is homogeneous with respect to all Z̃_{i,β,j}. Let V (M̄) = ∑_{p=1}^{r} v_{p} · B be the free module over B generated by v_{1}, …, v_{r}, then

(M̄; V (M̄); ν; , v_{1}, …, v_{r})

is a g.Lie module. We can define a homomorphism ϕ : M → L, which is the identity mapping on Q and its restriction to Ũ_{i,β,j} is the isomorphism onto W_{i,β}^{[0]}; then ϕ extends to a homomorphism τ from B to the envelope En(A) and, therefore, induces a homomorphism (ϕ, ψ) from the g.Lie module (M̄; V (M̄)) to A.

Let V_{0} denote the maximal submodule of V (M̄) over B homogeneous with respect to each Z_{i,β,j}; put

B = (M̄, V̄),  (3.8)

where V̄ = V (M̄)/V_{0}.

Lemma 3.2. T (B) ⊆ T (A).

Proof. Pick a multilinear f = f(x_{1}, …, x_{n}) ∈ T (A) and suppose f̄ ∉ V_{0} under some substitution x → x̄ ∈ M̄. Since f is multilinear, we may assume that all x̄_{l} are homogeneous with respect to each Z̃_{i,β,j}, hence f̄ is homogeneous too. Also,

ψ(f̄) = f(ϕ(x̄_{1}), …, ϕ(x̄_{n})) = 0

because f ∈ T (A); hence, f̄ must be in V_{0}. □

Lemma 3.3. V̄ is finite dimensional.

Proof. Consider the envelope En(B); we claim that it is finite dimensional. Indeed, let I_{0} be the maximal ideal in the kernel of the homomorphism τ : B → En(A) which is homogeneous with respect to all Z̃_{i,β,j}. Obviously, I_{0} is generated by all homogeneous elements from Ker(τ); also, ∑_{p+1}^{r} v_{p} · I_{0} ⊆ V_{0}, so it suffices to show that B/I_{0} is finite dimensional.

First, En(Q, ν) ⊆ B consists of elements of degree 0 with respect to all Z̃_{i,β,j}, hence Ker(τ)∩En(Q, ν) ⊆ I_{0} and En(Q, ν) is isomorphic to a subalgebra of En(A), in particular, it is of finite dimension.

Next, B is spanned by elements of the form

a_{0}b_{1}a_{1} … a_{p−1}b_{p}a_{p}  (3.9)

where a_{0}, …, a_{p} belong to En(Q, ν) and the elements b_{1}, …, b_{p} ∈ ν(Ũ) are homogeneous with respect to all Z̃_{i,β,j} and of degree ≥ 1 with respect to one of them. If p is greater then the nilpotency index of the radical of En(A), then element (3.9) lies in the kernel of τ and therefore it belongs to I_{0}. Hence modulo I_{0} the algebra B is the span of elements (3.9), where p is bounded, hence B is finite dimensional modulo I_{0}. □

By definition, the space V̄ is homogeneous with respect to all Z̃_{i,β,j}; let V_{1} denote the span of homogeneous elements with respect all Z̃_{i,β,j} and of degree ≥ 2 with respect to one of them. Then V_{1} is a submodule; put V^{ξ} = V̄ /V_{1}, L^{ξ} = M̄ and

A^{ξ} = (L^{ξ}, V^{ξ}) .

Recall that (3.7) gives a canonical decomposition of L^{ξ}; thanks to factorization modulo V_{1}, the finite dimensional g.Lie module A^{ξ} is linearized. Also, by definition we have (3.4), (3.5).

Obviously,

T (A) ⊆ Γ ∩ ⋂_{ξ∈J} T (A^{ξ}) ;

let Γ′ denote the right hand side and suppose that Γ′ ≠ T (A), then there exists a multilinear f = f(x_{1}, …, x_{n}) ∈ Γ′ ⊆ Γ and x̄ ∈ V(f, A). We will prove that f ∉ T (A^{ξ}) where ξ = ξ(x̄) (it will contradict to the choice of f).

To every x̄_{j} ∈ Spl(L) let us define x̃_{j} ∈ Spl(M̄):

(a) if x̄_{j} ∈ D_{i,α}, then α ∈ π_{i}(x̄) and therefore D_{i,α} ⊆ P_{i}; hence we can put x̃_{j} = x̄_{j};

(b) similarly, x̃_{j} = x̄_{j} if x̄_{j} ∈ C_{i};

(c) suppose x̄_{j} ∈ W_{i,β} and let m = m_{β}, then there are m variables x_{j_{1}}, …, x_{j_{m}} such that x̄_{j_{t}} ∈ W_{i,β}; we put x̃_{j_{t}} = x̄_{j_{t}} ∈ U_{i,β,j}.

The mapping x̃_{j} → x̄_{j} extends to a homomorphism (ϕ, ψ) from B to A, where ψ(f̃) = f̄ ≠ 0. Also, f̃ is homogeneous of degree ≤ 1 with respect to every Z̃_{i,β,j}, hence f̃ ∉ V_{1} and f does not vanish in V^{ξ} = V̄ /V_{1} under the substitution x_{i} → x̃_{i}. So, (3.3) holds and therefore lemma 3.1 is proven. □

We continue the proof of proposition 3.1. From (3.3) and lemma 2.15 it follows that there is a ”counter-example” T (A^{ξ}) ⊂ Γ′, where Γ′ ⊆ Γ + T (A^{ξ}) and

(Γ′)^{#} ⊆ Γ^{#} ⊆ Q(A) ⊆ Q(A^{ξ}) .

The g.Lie module A^{ξ} is linearized; let us show that

type(Γ, A^{ξ}) ≤ type(Γ, A) .  (3.10)

Take x̄ ∈ V(Γ, A) such that ξ = ξ(x̄); by (3.2) we have

∥a(A^{ξ}, Γ)∥ ≤ dim(D^{ξ}) = ∥a(x̄)∥ ≤ ∥a(A, Γ)∥

and

a(A^{ξ}, Γ) ≤ a(x̄) ≤ a(A, Γ) ,

without loss of generality we may assume that these are equalities. Similarly, |b(A^{ξ}, Γ)| ≤ |b(A, Γ)|, b(A^{ξ}, Γ) ≤ b(A, Γ) and ht(A^{ξ}, Γ) ≤ ht(A, Γ).

Recall that T (A) ⊆ T (B) ⊆ T (A^{ξ}) , hence, by lemma 2.19,

t(Γ, A^{ξ}) ≤ t(Γ, A) ,

hence, (3.10) holds.

Finally, let us show that there exists ȳ ∈ V(Γ′, A^{ξ}) such that

π(ȳ) = pr(D^{ξ}) , ω(ȳ) = pr(W^{ξ}) .

On the contrary, suppose it is untrue for all ȳ ∈ V(Γ′, A^{ξ}), then

∥a(ȳ)∥ < dim(D^{ξ}) = ∥a(x̄)∥ ≤ | a(Γ, A)∥

or

|b(ȳ)| < |pr(W^{ξ})| = |b(x̄)| ≤ |b(Γ, A)| .

Hence, type(Γ′, A^{ξ}) must be lesser than that of T (A) ⊂ Γ, which is impossible. So, replacing the given “counter-example” with T (A^{ξ}) ⊂ Γ′ we get proposition 3.1. □

2. Irreducible “counter-example”

First, we are going to prove that the types of Γ and Γ^{#} with respect to A are equal; to this end we need the following statement.

Lemma 3.4. Let T (A) ⊂ Γ be a ”counter-example” and let x̄ ∈ V(Γ^{#}, A) such that π(x̄) or ω(x̄) is not empty. Then there exists ȳ ∈ V(Γ, A) such that

π(x̄) = π(ȳ) , ω(x̄) = ω(ȳ) .  (3.11)

Proof. Let x̄ = (x̄_{1}, …, x̄_{n}) ∈ V(f, A) for some multilinear f ∈ Γ^{#}; By hypothesis, there is x̄_{j} lying in a simple component D_{i,α} or W_{i,β}.

Consider the first case; the commutator subalgebra of D_{i,α} is equal to D_{i,α}, hence x̄_{j} ∈ D_{i,α} can be represented as a Lie polynomial g = g(x_{n+1}, …, x_{m}) from [L_{i}⟨X⟩, L⟨X⟩] in some x̄_{n+1}, …, x̄_{m} ∈ D_{i,α},

x̄_{j} = g(x̄_{n+1}, …, x̄_{m}) .

Replacing x_{j} with g in f, we get f_{1} = f_{1} ∈ Γ and

f_{1}(ȳ) = f_{1}(x̄_{1}, …, x̄_{j−1}, x̄_{j+1}, …, x̄_{n}, x̄_{n+1}, …, x̄_{m}) = f(x̄) ≠ 0 .

We can assume that g is the sum of multilinear polynomials, hence, we can replace g by one of them to make f_{1} multilinear and therefore ȳ ∈ V(Γ, A). Equalities (3.11) hold by definition.

The case x̄_{j} ∈ W_{i,α} is similar. First, if W_{i,α} is a non-trivial D-module, then [W_{i,α}, D] = W_{i,α} and we make use the same argument as above. Otherwise, by lemma 2.5, W_{i,α} is contained in a weight subspace N_{i}(λ) with λ ≠ 0. Hence

x̄_{j} ∈ ∑_{t≥1}[W_{i,α}, \underbrace{C, …, C}_{t}]

and we can use the same argument. □

Corollary 3.1. Let T (A) ⊂ Γ be a ”counter-example”, then

type(Γ, A) = type(Γ^{#}, A) .

Proof. First, type(Γ, A) ≤ type(Γ^{#}, A) by (2.72). From the lemma above it folows that a(Γ^{#}, A) = a(Γ, A), b(Γ^{#}, A) = b(Γ, A) and ht(Γ^{#}, A) = ht(Γ, A). So, thanks to lemma 2.19, we need to show only that

t(Γ^{#}, A) ≤ t(Γ, A) .  (3.12)

Put a = a(Γ^{#}, A) and let b = t(Γ^{#}, A), we can assume that |b| ≠ 0. Then for all c ∈ ℕ (see Step 1 of the definition of (2.57))

S_{a,b,c+1}(Γ^{#}) ⊆ S_{a,b,c}(Γ)

which yields (3.12). □

Proposition 3.2. There exists a ”counter-example” T (A) ⊂ Γ of the smallest type such that A is a linearized g.Lie module and (3.2) holds for all x̄ ∈ V(A, Γ).

Proof. We need g.Lie submodules of A of a special form.

Take α ∈ pr(D); there is l ∈ {1, …, k} such that α ∈ pr(D_{l}); let L_{l}^{α} be the ideal of L generated by all D_{l,β}, β ≠ α and S(L_{l}). Also, put L_{i}^{α} = L_{i} provided i ≠ l and L^{α} = ∑_{i=1}^{k} L_{i}^{α}; then L^{α} = (L^{α}; L_{1}^{α}, …, L_{k}^{α}) is a k-sorted subalgebra of L. Consider the submodule A^{α} = (L^{α}; V^{α}) of A generated by L^{α}.

Lemma 3.5. For all α we have type(Γ, A^{α}) < type(Γ, A).

Proof. Recall there exists x̄ ∈ V(Γ, A) such that (3.2) takes place. Hence, we have ∥a(Γ, A)∥ = dim(D) and therefore

∥a(Γ, A^{α})∥ ≤ dim(D^{α}) < dim(D) = ∥a(Γ, A)∥ .

□ Now take β ∈ pr(W), then β ∈ pr(W_{l}) for some l; put

S^{β} = D_{l} ∪ ⋃_{γ≠β} W_{l,γ} ∪ W_{l}^{[1]} ∪ C_{l}  (3.13)

and let M^{β} denote the k-sorted subalgebra of L generated by S^{α} ⊆ L_{l} and Gn_{i}(L), i ≠ l. We are going to find a canonical decomposition for M^{β}; for all i put

I_{i}^{β} = I_{i}(M^{β}) = ∑_{j≠i}[M_{i}^{β}, M_{j}^{β}] .

Take i ≠ l and let M_{i} be the sum of I_{i}^{β} and the subalgebra generated by Gn_{i}(L); then M_{i} is an ideal of M^{β}, hence M_{i}^{β} = M_{i}. By lemma 2.6 , Gn_{i}(L) generates the algebra L_{i} modulo I_{i}, hence we can put Gn_{i}(M^{β}) = Gn_{i}(L). Actually, D_{i} is a maximal semisimple subalgebra of M_{i}^{β}, hence we can put D_{i}(M^{β}) = D_{i}. Modulo I_{i}^{β} the solvable radical S(M_{i}^{β}) is equal to that of ⟨Gn_{i}(L)⟩; by definition

N_{i}(M^{β}) = N(M^{β}) + I_{i}^{β} = [M_{i}^{β}, S(M_{i}^{β})] + I_{i}^{β} ,

hence, it contains (C_{i}, C) and, by lemma 2.5, W_{i}^{[0]}. Hence, S(M_{i}^{β}) = N_{i}(M^{β}) + C_{i}; besides, N_{i}(M^{β}) ⊆ N_{i}, hence the sum is direct, so we may set C_{i}(M^{β}) = C_{i}.

Also, by definition

M_{i}(M^{β}) = N_{i}(M^{β})^{[1]} + N_{i}(M^{β})^{2} + ⟨C_{i}⟩^{2} + I_{i}^{β} ,

hence, attaching W_{i}, we get N_{i}(M^{β}); besides M_{i}(M^{β}) lies in M_{i}, hence we can put W_{i}(M) = W_{i}.

Now consider M_{l}; let J_{0} denote the subalgebra generated by

∑_{γ≠β} W_{l,γ} + W_{l}^{[1]} ,

then S′ = ⟨C_{l}⟩ + J_{0} + I_{l}^{β} is an ideal of M^{β} (it is closed under multiplication by S^{β} and Gn_{i}(L)). Moreover, it is solvable and J + D_{l} = M_{l}^{β}, hence, D_{l}(M^{β}) = D_{l} and S′ is the solvable radical of M_{l}^{β},

S(M_{l}^{β}) = ⟨C_{l}⟩ + J_{0} + I_{l}^{β} .

Next, J_{0} ⊆ N(M_{l}^{β}) by lemma 2.5, hence N_{l}(M^{β}) includes

N′ = ⟨C_{i}⟩^{2} + J_{0} + I_{l}^{β} ;

besides, N′ + C_{i} = N_{l}(M^{β}) + C_{i} = S(M_{l}^{β}) and N_{l}(M^{β}) ⊆ N_{l}, hence, N_{l}(M^{β}) = N′ and we can put C_{l}(M^{β}) = C_{l}.

Now, let us find W_{l}(M^{β}); by definition it is a D-module completing M_{l}(M^{β}) to N_{l}(M^{β}) with simple components lying in the weight subspaces with respect to ⟨C⟩. Recall that

M_{l}(M^{β}) = N_{l}(M^{β})^{[1]} + N_{l}(M^{β})^{2} + ⟨C_{l}⟩^{2} + I_{l}^{β} ,

obviously it lies in M_{l}. We choose W_{l}(M^{β}) as the sum of two parts: the complement M_{l}(M^{β}) to N_{l}(M^{β}) ∩ M_{l} and that of N_{l}(M^{β}) ∩ M_{l} to N_{l}(M^{β}).

Clearly, N_{l}(M^{β}) ∩ M_{l} includes

M′ = W_{l}^{[1]} + J_{0}^{2} + ⟨C_{l}⟩^{2} + I_{l}^{β} ;

on the other hand,

M′ + ∑_{γ≠β} W_{l,γ} = J_{0} + ⟨C_{l}⟩^{2} + I_{l}^{β} = N_{l}(M^{β}) ,

hence,

∑_{γ≠β} W_{l,γ}  (3.14)

is a complement of N_{l}(M^{β}) ∩ M_{l} to N_{l}(M^{β}).

Next, all W_{l,γ}^{[1]} lie in W_{l}(M^{β})^{[1]}, hence, the complement M_{l}(M^{β}) to N_{l}(M^{β}) ∩ M_{l} is the direct sum of several D-submodules of W_{l,β}^{{1}}; they have the form

W_{l,β} · (ad(c) − λ(c))  (3.15)

for suitable c ∈ C and λ ∈ Λ_{l}, cf. (2.29). In particular, they are isomorphic to W_{l,β} as M^{β}-modules, but their height is less than that of W_{l,β}.

Lemma 3.6. Let B^{β} = (M^{β}; V^{β}) be the g.Lie submodule of A generated by M^{β}, then type(Γ, B^{β}) < type(Γ, A) .

Proof. On the contrary, suppose it is untrue; since D(M^{β}) = D,

∥a(Γ, B^{β})∥ ≤ dim(D(M^{β})) ≤ ∥a(Γ, A)∥ ,

hence, by hypothesis this must be an equality. Similarly, a(Γ, B^{β}) = a(Γ, A) and by lemma 2.19, t(Γ, B^{β}) = t(Γ, A).

Next, let us show that |b(Γ, B^{β})| ≤ |b(Γ, A)| and b(Γ, B^{β}) ≤ b(Γ, A). Pick x̄ ∈ V(f, B^{β}), where f ∈ Γ; each x̄_{j} is equal to a Lie polynomial h_{j} in some elements from Spl(L) (cf. lemma 2.17). Notice that Spl(L_{i}) = Spl(M_{i}^{β}) unless i = l, hence , h_{j} = x̄_{j}. From the description of Gn_{l}(M^{β}) given above the same holds for x_{j} ∈ X_{l}, unless x̄_{j} belongs to a module (3.15). Otherwise, h_{j} = [ȳ_{j}, ȳ′_{j}] − λ · ȳ_{j} where ȳ_{j} ∈ W^{β}, ȳ′_{j} ∈ C, and λ = λ(ȳ′_{j}). Hence, f(x̄) is equal to a sum of elements of the form g(ȳ), where g is multilinear from Γ, ȳ ∈ V(Γ, A) and

b^{(m,n)}(ȳ) = b^{(m,n)}(x̄)

for all m, n. Hence, b(ȳ) = b(x̄) = b(Γ, A) thanks to (3.2); therefore

b(Γ, A) = b(Γ, B^{β}) .

Finally, by (3.2) there is ȳ_{j} lying in W_{l,β}, hence one of x̄_{l} is from a module (3.15) whose height is less than that of W_{l,β}, hence ht(x̄) < ht(Γ, A). □

We continue proving proposition 3.2. Put Υ = Γ ∩ Ψ, where

Ψ = ( ⋂_{α∈pr(D)} T (A^{α})) ∩ ( ⋂_{β∈pr(W)} T (B^{β})) ;

we claim that Υ is critical with Υ^{#} ⊆ Ψ. Actually, if it were untrue, then by lemma 2.15 there would exist a “counter example” of the form T (A^{α}) ⊆ Γ′, where Γ′ ⊆ Γ + T (A^{α}) and (Γ′)^{#} ⊆ Γ^{#}, or T (B^{β}) ⊆ Γ′, where Γ′ ⊆ Γ + T (B^{β}) and (Γ′)^{#} ⊆ Γ^{#}. From lemma 3.5 and 3.6 it follows that its type is less than type(Γ, A) which is impossible.

So, we can assume that Γ^{#} ⊆ Ψ; take x̄ ∈ V(Γ^{#}, A) and suppose that (3.2) does not hold. First, if π(x̄) ≠ pr(D), then there is α ∈ pr(D)\π(x̄) and x̄ ∈ V(Γ, A^{α}), which contradicts to the hypothesis Γ ⊆ Ψ. Similarly, if ω(x̄) ≠ pr(W), then x̄ ∈ V(Γ, B^{β}) for β ∈ pr(W)\ω(x̄). □

From (3.2) and lemma 3.4 it follows that

a(x̄) = a(Γ, A) , b(x̄) = b(Γ, A) , ht(x̄) = ht(Γ, A)  (3.16)

for all x̄ ∈ V(Γ^{#}, A).

Let L be a k-sorted Lie algebra with a cononical decomposition; put

J_{i} = J_{i}(L) = W_{i}^{[1]} + ∑_{t≥1}[W_{i}^{[0]}, \underbrace{W^{[0]}, …, W^{[0]}}_{t}] + I_{i} ;  (3.17)

it is an ideal of L; by lemma 2.7,

L_{i} = D_{i} + ⟨C_{i}⟩ + W_{i} + J_{i} .

Proposition 3.3. There exists a “counter-example T (A) ⊆ Γ of the smallest type such that

(1) A is linearized ;

(2) relations (3.2) hold for all x̄ ∈ V(Γ, A) ;

(3) Let f(x) = f(x_{1}, …, x_{n}) ∈ Γ^{#} be multilinear; take some elements x̄_{j} ∈ ⋃_{i=1}^{k} L_{i} where at least one of these, say x̄_{p}, belongs to J_{i} for some i. Then f(x̄) = 0.

Proof. Note that we have got already a “counter-example” of the smallest type satisfying (1) and (2). Take β ∈ pr(W), then β ∈ pr(W_{l}) for some l. We define a submodule C^{β} = (N^{β}; U^{β}) of A, where N^{β} is the k-sorted sublagebra of L generated by Gn_{i}(L), where i ≠ l, and a subset T^{β} ⊆ L_{l}, that is the union of D_{l}, C_{l},

⋃_{γ≠β} W_{l,γ}  (3.18)

and

⋃_{t≥1}[W_{l,β}^{[0]}, \underbrace{W_{l}^{[0]}, …, W_{l}^{[0]}}_{t}] .  (3.19)

Like in lemma 3.6 one can define a canonical decomposition such that Gn_{i}(ℕ^{β}) = Gn_{i}(L) provided i ≠ l. Next, put I_{l}^{β} = I_{l}(ℕ^{β}); clearly, D_{l} is a maximal semisimple subalgebra of N_{l}^{β} and the radical, S(N_{l}^{β}) is the sum of ⟨C_{l}⟩, (3.18), (3.19) and I_{l}^{β}.

Next, N_{l}(N^{β}) contains (3.18) by lemma 2.5; also, (3.19) is invariant under the action by ⟨C⟩, hence all nonzero weight subspaces of (3.19) are contained in N_{l}(N^{β}). Finally, the zero weight subspace of (3.19)is the direct sum of irreducible D-submodules; all non-trivial ones lie in N_{l}(N^{β}), hence a complement of N_{l}(N^{β}) to S(N_{l}^{β}) can be chosen as the sum of C_{l} (because N_{l}(N^{β}) ⊆ N_{l}) and a trivial D-submodule C′_{l} from the zero weight subspace of (3.19). Obviously, C_{l} + C′_{l} and ∑_{i≠l} C_{i} generate a nilpotent subalgebra commuting with D, hence we may put C_{l}(N^{β}) = C_{l} + C′_{l}.

Now, M_{l}(N^{β}) is contained in M_{l}(L) and includes all W_{l,γ}^{[1]} for γ ≠ β, hence, in order to complete it to N_{l}(N^{β}) we take the the sum of (3.18) and a number of irreducible Dsubmodules from weight subspaces of (3.19) with respect to ⟨C(N^{β})⟩ (as a matter of fact, C′_{l} ⊆ N_{l} acts nilpotently, hence they are just weight subspaces with respect to ⟨C⟩).

Now we can prove an analog of the lemma above for C^{β}.

Lemma 3.7. For all β type(Γ, C^{β}) < type(Γ, A) .

Proof. The argument will be similar to the proof of lemma 3.6. We suppose that the statement is untrue; in absolutely the same way one can show that ∥a(Γ, C^{β})∥ = ∥a(Γ, A)∥, a(Γ, C^{β}) = a(Γ, A) and t(Γ, C^{β}) = t(Γ, A).

Next, take x̄ = (x̄_{1}, …, x̄_{n}) ∈ V(f, C^{β}), where f ∈ Γ; then x̄_{j} = h_{j}, where h_{j} is a Lie polynomial in elements from Gn_{i}(L). Moreover, h_{j} = x̄_{j} unless x_{j} ∈ X_{l}, otherwise x̄_{j} belongs to a simple component of W_{i}(N^{β}) from (3.19). In the latter case h_{j} is a sum of multilinear Lie polynomials in elements from Gn_{i}(L), where at least two of them belong to W_{l} (moreover, one of them must be in W_{l,β}). Hence, f(x̄) is equal to a sum of elements of the form g(ȳ), where g ∈ Γ, ȳ ∈ V(Γ, A), where

|b(x̄)| + 1 ≤ |b(ȳ)| .

Therefore |b(Γ, C^{β})| < |b(Γ, A)|, and the lemma is proven. □

Now put

Υ = Γ ∩ ⋂_{β∈pr(W)} T (B^{β}) ∩ T (C^{β}) ,  (3.20)

by lemma 2.15 Υ must be a critical T-ideal with an apropriate sequence contained in all T (C^{β}) and T (B^{β}) (otherwise, thanks to lemma 3.6 and 3.7, we would have a “counterexample” of lesser type).

So, we can assume that Γ coincides with Υ and

Γ^{#} ⊆ ⋂_{β∈pr(W)} T (B^{β}) ∩ T (C^{β}) ,

observe that this replacement is harmless for properties (1), (2), so we need only to check (3).

Suppose f(x̄) ≠ 0; arguing like in lemma 2.17, we may replace all x̄_{j}, j ≠ p, with Lie polynomials in elements from Spl(L). Hence we may assume that all x̄_{j} ∈ Spl(L) unless j = p. Moreover, since f is linear in x_{p}, the element x̄_{p} can be taken either from W_{i}^{[1]} or

[W_{i}, \underbrace{W^{[0]}, …, W^{[0]}}_{t}] ,  (3.21)

where t ≥ 1.

Consider the first case; recall that W_{i}^{[1]} is the sum of subspaces

W_{i,β} · (λ(c_{1}) − ad(c_{1})) … (λ(c_{p}) − ad(c_{q}))

where q ≥ 1 and c_{1}, …, c_{q} ∈ C_{i}, see (2.29); hence, we may assume that x̄_{p} belongs to such a subspace. Since A is linearized, there are no more x̄_{j} from W_{i,β}; hence, all x̄_{j}, including j = p, are elements in M^{β}. It is impossible, because Γ ⊆ T (B^{β}) and f(x̄) ≠ 0. Hence, x̄_{p} lies in (3.21), and we may assume that it is an element (3.19) for some β ∈ pr(W_{i}); hence, all x̄_{j} belongs to N^{β} and f(x̄) must vanish. □

Annihilators.

We are going to minimize the set of values taken by polynomials in Γ^{#} under all possible substitutions.

Lemma 3.8. There exists a ”counter-example” T (A) ⊂ Γ of the smallest type, satisfying (1)-(3) from proposition 3.3 and the following condition. Let A = (L, V), then the verbal submodule

Γ^{#}[A] = {f(x̄)| f = f(x_{1}, …, x_{n}) ∈ Γ^{#}; x̄_{j} ∈ L}

is an irreducible submodule over En(A) and contained in any nonzero submodule of V.

Proof. Put V_{1} = Γ^{#}[A]; we shall carry out the induction on dim(V_{1}). Consider the set of all nonzero submodules of V which do not contain V_{1}; if it is empty, then V_{1} is irreducible and the lemma holds. Otherwise, take a maximal element from this set, V_{0}, and consider the quotient A′ = (L; V/V_{0}).

First, suppose that Υ = Γ ∩ T (A′) has no an appropriate sequence in T (A′), then, by lemma 2.3, Ψ = Γ + T (A′) is critical with the appropriate sequence in Γ^{#}, so Ψ^{#} = Γ^{#} +T (A′). Observe that the first component of A, the k-sorted Lie algebra L is just that of A′, hence T (A′) ⊂ Ψ satisfies (1)-(3) from proposition 3.3 and its type is the smallest one. Also, thanks to the maximality of V_{0}, the module Ψ^{#}[A′] = Γ^{#}[A]/V_{0} = V_{1} + V_{0}/V_{0} is irreducible and any nonzero submodule of V/V_{0} contains it.

Thus, we may assume that Υ is critical with Υ^{#} ⊆ T (A′), hence, T (A) ⊂ Υ is a”counter-example” of the smallest type satisfying (1)-(3) and Υ^{#}[A] ⊆ V_{0} ∩ Γ^{#}[A] ≠ V_{1}, hence, we can use the induction hypothesis. □

Thus, we assume that the ”counter-example” satisfies the hypothesis of the lemma, we call it an irreducible one.

Let M ⊆ L, then the set of all v ∈ V

v · ν(M) = 0

is called the annihilator of M and denoted by Ann(M). Obviously, it is a subspace of V.

Lemma 3.9. The verbal submodule Γ^{#}[A] is contained in Ann(W).

Proof. Suppose it is untrue, then

g(x̄_{1}, …, x̄_{n+1}) = f(x̄) · μ(x̄_{n+1}) ≠ 0

for some x̄, μ ∈ {ρ_{l}, δ_{l}} and x̄_{n+1} ∈ W. We may assume that f is multilinear and x̄ ∈ V(Γ^{#}, A). Hence, g is a multilinear one from Γ and x̃ = (x̄_{1}, …, x̄_{n+1}) is in V(Γ, A). Since x̄_{n+1} ∈ W, |b_{i}(x̄)| = |b_{i}(x̃)| − 1, which contradicts to (2) of proposition 3.3. □

As a matter of fact, Γ^{#}[A] annihilates the ideal of L generated by W; actually, Γ^{#}[A] is a submodule of V over En(A), hence, if a ∈ W and b ∈ L, then by (2.10), (2.11) the operators ρ_{j}([a, b]), δ_{j}([a, b]) act trivially on Γ[A].

Lemma 3.10. Let α ∈ pr(D_{i}) for some i, then Ann(D_{i,α}) ∩ Γ^{#}[A] is a submodule of V over En(L, ν).

Proof. Let v ∈ Ann(D_{i,α}) ∩ Γ^{#}[A], we need to show that for all a ∈ L and b ∈ D_{i,α}

v · μ(a)μ′(b) = 0  (3.22)

where μ ∈ {ρ_{l}, δ_{l}}, μ′ ∈ {ρ_{l′}, δ_{l′}} First all, if l ≠ l′, then it is trivial by (2.12). So, μ, μ′ ∈ {ρ_{l}, δ_{l}} for some l.

We may assume that a ∈ D_{β} + ⟨C⟩ for some β ≠ α, because Γ^{#}[A] is contained in the annihilator of the ideal generated by W. Hence, a commutes with D_{i,α} and, thanks to (2.10), (2.11), we can swap μ(a) and μ′(b) and get (3.22). □.

Take α ∈ pr(D_{i}) and put

Ann(α) = Ann(D_{i,α}) ∩ Γ^{#}[A] ;

for Ω ⊆ pr(D) put let

Ann(Ω) = ⋂_{α∈Ω} Ann(α) ;

it is a submodule of V over En(A). Now let Ω = Ω(Γ, A) denote the set of all α ∈ pr(D) such that Ann(α) ≠ 0; by lemma 3.8, Γ^{#}[A] is contained in every Ann(α), hence

Γ^{#}[A] ⊆ Ann(Ω) .  (3.23)

3. Finite dimensional embedding

In this section we consider the case

b(Γ, A) ≤ t(Γ, A) .

Proposition 3.4. Let Υ denote the T-ideal (2.57) where, a = a(Γ, A), b ≥ b(Γ, A) and c is the nilpotency index of ⟨C⟩. Then there exists g.Lie module A′ such that T (A′) = Υ + T (A) and A′ is finite dimensional over a field extension of F.

This statement is an analog of proposition 1.6. At the first stage we prove that Υ is the ideal of identities of a g.Lie module admitting a scalar extension by an associative commutaive affine algebra K, so the resulting g.Lie module satisfies certain finitness conditions. For we need some technical facts describing substitution x → x̄ under which the typical generators of (2.57) do not vanish.

First, take a multilinear f = f(x) = f(x_{1}, …, x_{n}) ∈ Γ that is not an identity of A. Since it is multilinear, f(x̄) ≠ 0 under some substitution x → x̄, where

x̄ ∈ Spl_{i}(L) ∪ ⟨C_{i}⟩^{2} ∪ J_{i}

provided x ∈ X_{i}; moreover, by proposition 3.3 x̄ ∈ Spl_{i}^{+}(L), where

Spl_{i}^{+}(L) = Spl_{i}(L) ∪ ⟨C_{i}⟩^{2} .  (3.24)

This substitution x̄ looks almost like an element in V(Γ, A), the only difference is that some entries x̄ may be in ⟨C_{i}⟩^{2}. However, it is not important, when we count only representatives from simple components of D and W. Thus, we can define the sets π(x̄), ω(x̄) like for elements of V(Γ, A), see (3.1); also, (3.2) holds for such x̄. Indeed, the variable x with x̄ ∈ ⟨C_{i}⟩^{2} can be replaced with a Lie polynomial from L⟨X_{i}⟩ in elements from C_{i}, so, x̄ can be replaced with ȳ ∈ V(Γ, A) with π(x̄) = π(ȳ) and ω(x̄) = ω(ȳ).

Now let Γ′ denote T-ideal (2.57), where a = a(Γ, A), b ≥ b(Γ, A) and ⟨C⟩^{c} = 0. Suppose Γ ⊈ T (A); let h(x) = h(x_{1}, …, x_{n}) be a typical generator (2.66) of Γ′ and let x → x̄ be a substitution such that h(x̄) ≠ 0 and all x̄_{j} ∈ Spl^{+}(L), where

Spl^{+}(L) = ⋃_{i=1}^{k} Spl^{+}(L_{i}).

The next statement follows immediately from the multiplication table for components of Spl_{+}(L_{i}).

Lemma 3.11. Let ϕ ∈ L⟨X_{i}⟩ and let x → x̄ be a substitution where x̄ ∈ Spl_{i}^{+}(L) and ϕ̄ ∉ J_{i}. Then we have the following:

(1) ϕ̄ ∈ ⟨C_{i}⟩ + J_{i} iff x̄ ∈ ⟨C_{i}⟩ for all x ∈ varb(ϕ);

(2) ϕ̄ ∈ D_{i,α} + J_{i} iff x̄ ∈ D_{i,α} for all x ∈ varb(ϕ).

(3) ϕ̄_{x} ∈ W_{i,β} + J_{i} iff one variable from varb(ϕ) takes value in W_{i,β} and the others are in D_{i} ∪ C_{i} (note that [W_{i}, ⟨C_{i}⟩^{2}] ⊆ J_{i}).

Consider the Lie polynomials ϕ_{x} defined in (2.63), (2.64); under x → x̄ they takes values in Spl_{+}(L_{i}) modulo J_{i}, provided x ∈ X_{i}. Moreover, deg(ϕ_{x}) ≥ c, hence, under x → x̄ ϕ_{x} takes value in Spl(L_{i})\⟨C_{i}⟩ (modulo J_{i}), i.e., in simple components of D_{i} and W_{i} (modulo J_{i}):

ϕ̄_{x} ∈ ( ⋃_{α∈pr(D_{i})} D_{i,α}) ∪ ( ⋃_{β∈pr(W_{i})} W_{i,β})  (3.25)

Like in the associative case, we can describe the location of variables x_{j} with x̄_{j} in standard and non-standard submodules of W_{i}. In fact, we will get even more general statement showing in particular that t(Γ, A) ≤ b(Γ, A) (see corollary 3.2). Here we use notations of the definition of (2.66); besides, put J = ∑_{i=1}^{k} J_{i} .

Lemma 3.12. Suppose that Γ′ ⊈ T (A), then

b = b(Γ, A) .  (3.26)

Moreover, let h be a typical generator (2.66) of Γ′ and suppose h̄ ≠ 0 under x → x̄ ∈ Spl_{+}(L), then for all m, l ∈ ℕ the following statements hold: Q_{m,l}: take p ∈ η^{(m,l)} and let y_{p} denote the fixed variable in Θ_{p}; then modulo J the value

of ψ_{y_{p}} under x → x̄ belongs to a non-standard L-submodule W_{β} of dimension l satisfying (2.49);

Q_{m,0}: for all p ∈ η^{(m,0)} there is a variable x ∈ Θ_{p} such that modulo J the value of ϕ_{x} under x → x̄ belongs to a standard L -submodule W_{β} satisfying (2.49).

Proof. Put

ω^{(m,l)} = {β ∈ pr(W)| W_{β} is non − standard, dim(ad_{β}(L)) = m;

dim(W_{β}) = l} ,
ω^{(m,0)} = {β ∈ pr(W)| W_{β} is standard, dim(ad_{β}(L)) = m} ;

by (3.2), |ω^{(m,l)}| = b^{(m,l)}(x̄) for all m, l, hence, equality (3.26) will follow from

b^{(m,l)} = b^{(m,l)}(x̄) .  (3.27)

To prove it and Q_{m,l} we carry out the induction on (m, l) with respect to the lexicographical order, where (m, l) < (m′, l′) if m < m′ or m = m′ and l < l′.

The initial step. Since b is a finite sequence, there exists m_{0} such that b^{(m,l)} = 0 for all (m, l) such that m ≥ m_{0}. By hypothesis, b(Γ, A) ≤ b, hence, b^{(m,l)}(x̄) ≤ b^{(m,l)} = 0 for such m and (3.27) takes place. Also, η^{(m,l)} is empty, hence Q_{m,l} trivially holds.

Now, suppose that Q_{m,l} and (3.27) are true for all m′ > m, we need to check these for m = m′ and for all l. There exists l_{0} such that b^{(m,l)} = 0 for all l ≥ l_{0}; since b^{(m′,l)}(x̄) = b^{(m′,l)} for all l and b(Γ, A) ≤ b, we get b^{(m,l)}(x̄) = b^{(m,l)} for all l ≥ l_{0}. Since η^{(m,l)} = ∅, we get Q_{m,l} too.

So, we can suppose that Q_{m,l} and ( 3.27) hold for all (m′, l′) > (m, l), we need to check these for (m, l). By the induction hypothesis, for all p ∈ η^{(m′,l′)} there exists x ∈ Θ_{p} such that ϕ̄_{x} ∈ W_{β} for a suitable β (hereafter in the proof it means a relation modulo J). Also, by the induction hypothesis, b^{(m′,l′)}(x̄) = b^{(m′,l′)}, hence b^{(m′,l′)}(x̄) = |η^{(m′,l′)}| and the total number of such x-s is equal to

∑_{(m′,l′)>(m,l)} |η^{(m′,l′)}| = ∑_{(m′,l′)>(m,l)} b^{(m′,l′)}(x̄) = ∑_{(m′,l′)>(m,l)} |ω^{(m′,l′)}| ,

hence, for every β ∈ ω^{(m′,l′)} such x exists. Hence, by lemma 3.11, for all β ∈ ω^{(m′,l′)} there exists p and x ∈ Θ_{p} such that ϕ̄_{x} ∈ W_{β}.

Recall that A is linearized, hence, if ϕ̄_{x} ∈ W_{γ} for some x ∈ Θ_{p} and p ∈ η^{(m,l)}, then

γ ∉ ω^{(m′,l′)}  (3.28)

for all (m′, l′) > (m, l).

Pick p ∈ η^{(m,l)}; also, let y_{p} denote the fixed variable of Θ_{p}. By definition, the set Θ_{p}\{y_{p}} is broken down into a union of disjoint subsets Θ_{p}^{t}, t ≥ 1, where Θ_{p}^{t} contains a^{(t)} · t elements and for every x ∈ Θ_{p}^{t} the polynomial ϕ_{x} has form (2.63) (m is replaced by t).

Lemma 3.13. If t > m, then ϕ̄_{x} belongs to a simple component of D of dimension > m.

Proof. Suppose first that ϕ̄_{x} lies in a simple component D_{α} of dimension ≤ m, hence by lemma 3.11, ȳ ∈ D_{α} for all y ∈ varb(ϕ_{x}). Recall that ψ_{x} ∈ LCap_{t,1} (here we make use the notation of (2.63)), where t ≥ m + 1, hence ϕ̄_{x} must be equal to 0.

So, we have ϕ̄_{x} ∈ W_{β} for some β; by lemma 3.11 there is only one variable z ∈ varb(ϕ_{x}) such that z ∈ W_{β} and the others are from D ∪ C. By (3.28) dim(ad_{β}(L)) ≤ m < t; since ψ_{x} ∈ LCap_{t,1}, the variable z must be in varb(ψ_{x}), hence ψ̄_{x} ∈ W_{β}. Therefore, all variables in varb(ψ′_{x}) take value in D ∪ C, we may regard them as adjoint operators from ad(L) acting on W_{β}. But ψ′_{x} is skew-symmetric in t variables, hence ψ̄′_{x} must be equal to 0. □

Put

Y_{m} = ⋃_{t>m} Θ_{p}^{t} ,

for all x ∈ Y_{m} the element ϕ̄_{x} is in a simple component D_{α} of dimension > m. Next, the order of Y_{m} is equal to

∑_{t>m} |Θ_{p}^{t}| = ∑_{t>m} a^{(t)} · t ,

which is the total dimension of all simple components D_{α} of dimension > m. Denote by D the sum of these; recall the polynomial h_{0} (see Step 2 of the definition of (2.66)) is skew-symmetric with respect to the elements in Θ_{p}, hence, {ϕ̄_{y}| y ∈ Y_{m}} is a basis of the vector space D. So, if ϕ̄_{y_{p}} lies in a simple subalgebra D_{α}, then its dimension must be ≤ m.

Now, suppose l ≥ 1, then the polynomial ϕ_{x} for x = y_{p} has form (2.64). In particular, it belongs to LCap_{m+1,1}, hence its value must be in some W_{β}. If it were standard, then its dimension would be ≥ m + 1, which is impossible by (3.28). Hence, it is a non-standard one of dimension m; the value of ψ′_{y_{p}} belongs to W_{β}, hence Ē_{1}, …, Ē_{l^{2}} are operators from End(W_{β}). Since ϕ_{y_{p}} is skew-symmetric with respect to these operators, the dimension of W_{β} must be ≥ l. On the other hand, it does not exceed l thanks to (3.28), therefore, Q_{m,l} is proven in case l > 0. Besides, if b^{(m,l)}(x̄) < b^{(m,l)} = |η^{(m,l)}|, then there are p, p′ ∈ η^{(m,l)} such that the values of ϕ_{y_{p}} and ϕ_{y_{p′}} belong to the same simple component of W, which is impossible, because A is linearized.

Now suppose l = 0; consider the set Θ_{p}^{m} ∪ {y_{p}} and pick one of its elements, x. If ϕ̄_{x} ∈ D_{α}, then the dimension of D_{α} must be ≤ m. Indeed, if it were greater than m, then ϕ̄_{x} would be in D; on the other hand, h_{0} is skew-symmetric with respect to Θ_{p} which contains Y_{m} and {ϕ̄_{x}| x ∈ Y_{m}} is a basis of D, hence h would vanish.

So, the dimension of D_{α} is equal to m; denote by D′ the sum of all such simple parts of D. Observe that the total number of elements in Θ_{p}^{m} ∪ {y_{p}} is equal to |a^{m}| · m + 1, it is precisely dim(D′) + 1, hence there is at least one x ∈ Θ_{p}^{m} ∪ {y_{p}} such that ϕ̄_{x} lies in W_{i,β} for some β.

Suppose that dim(ad_{β}(L)) < m, then , since ϕ_{x} = ψ_{x} · ψ′_{x} ∈ LCap_{m,1}^{2}, ϕ̄_{x} must be equal to 0. Hence, the dimension of ad_{β}(L) is ≥ m; by (3.28) it is equal to m and W_{β} is standard; like in case l > 0 it implies immediately (3.26). Lemma 3.12 is proven. □

The lemma gives immediately the following.

Corollary 3.2. Let T (A) ⊂ Γ be an irreducible ”counter-example” of the smallest type, then

t(Γ, A) ≤ b(Γ, A) .

We continue the proof of proposition 3.4.

Let W be a unital right module over an associative algebra R, 1 ∈ R. It is said to be representable, if there exists a finite dimensional vector space W′ over a field extension H of F such that

(a) W′ is a unital right module over R_{H} = H ⊗_{F} R;

(b) W is embedded into W′ as a module over R.

The next statement is due to A.Anan’in [3].

Proposition 3.5. Let W be a finitely generated module over a finitely generated associative PI-algebra R satisfying a.c.c. for right ideals. Then W is a representable.

Consider F(A) = (F(L); F(V, L)), the g.Lie module of generic elements of A, see (2.18), and put R = En(F(A)). Recall that F(V, L) = V ⟨X⟩/T (A) is contained in the space of polynomial mappings Pol(L⃗; V) ≃ F [L⃗] ⊗_{F} V. Put

Υ_{1} = Υ + T (A)  (3.29)

where Υ is the T-ideal in the hypothesis of proposition 3.4). Let Υ̃_{1} = Υ_{1}/T (A), it is a submodule of F(V; L) over R.

By proposition 2.2, Π(L) is integral over tr_{U}[L], wherte V ∈ Q(U). By some technical reason we replace V with the following module; let d_{β} denote the dimension of W_{β}, put

U = D ⊕ ⨁_{β} d_{β}W_{β} ,

where β runs over all projectors in pr(W) on non-standard submodules. Clearly, every standard submodule is embedded into U, hence V is embedded into the sum of several copies of U, hence, V ∈ Q(U). Hence, by proposition 2.2, Π(L) is integral over K = tr_{U}(L). We can represent U in the following form. Let β ∈ pr(W), since the module W_{β} is irreducible, the algebra A_{β} = End(W_{β}) is equal to the envelope of ad_{β}(L). We consider A_{β} as the regular right module; it is isomorphic to the direct sum of d_{β} copies of W_{β} regarded as a right module over A_{β}. Hence, A_{β} ≃ d_{β}W_{β} as L-modules and

U ≃ D ⊕ ⨁_{β} A_{β} .  (3.30)

Lemma 3.14. The module Υ̃_{1} is invariant under the multiplication by elements in K.

Proof. Recall that K is generated by elements tr_{U}(E) where E is the product of operators ad(x), x ∈ X. To prove the lemma it suffices to show that for all h ∈ Υ̃ there exists h′ ∈ Υ such that h · tr_{U}(E) = h′ under any substitution of elements from L.

Take a typical generator h of Υ of form (2.66) and let p ∈ η^{(m,l)}, where l ≥ 1, and let y_{p} denote the fixed variable in Θ_{p}. Recall the polynomial ϕ_{y_{p}} has form (2.64); put

ϕ_{y_{p}} ∗ tr(E) = ∑_{j=1}^{l^{2}} ϕ_{y_{p}}|_{E_{j}·E} ,

note it is a polynomial of the same form as ϕ_{y_{p}}. Next, let h ∗_{m,l} tr(E) denote the sum ∑_{p} h_{p}, where p runs over η^{(m,l)} and h_{p} is obtained from h by replacing ϕ_{y_{p}} with ϕ_{y_{p}} ∗tr(E); then h ∗_{m,l} tr(E) ∈ Υ.

Also, let h ∗_{0} tr(E) denote the sum of polynomials h_{x}, x ∈ Θ_{0}, obtained from h under the substitution ϕ_{x}E in place of ϕ_{x}. Observe that ϕ_{x}E has form (2.63), hence h_{x} is still a typical generator of Υ. Finally put

h′ = h ∗_{0} tr(E) + ∑_{m,l≥1} h ∗_{m,l} tr(E) ,

where (m, l) runs over all parameters such that b^{(m,l)} ≠ 0; obviously, h′ ∈ Υ.

Lemma 3.15. The equality

h · tr_{U_{i}}(E) = h′  (3.31)

holds in Pol(L⃗; V).

Proof. We may assume that E is multilinear and independent from variables in varb(h), so h = h(x) = h(x_{1}, …, x_{n}), E = E(x_{n+1}, …, x_{q}) and therefore h′ = h′(y), where y = (x_{1}, …, x_{q}). Let us replace the variables x_{j} with some x̄_{j} in L; since the both sides are multilinear, we may assume that all x̄_{j} ∈ Spl^{+}(L) ∪ J.

In addition, suppose that h̄ = h(x̄) ≠ 0 and tr_{U}(E) ≠ 0, then for all x ∈ varb(h) the element x̄ belongs to Spl^{+}(L). Besides, since tr_{U}(Ē) ≠ 0, we have x̄_{j} ∈ D ∪ C for all j ∈ {m + 1, …, n}. By lemma 3.12, for all m, l ∈ ℕ and all p ∈ η^{(m,l)} the element ϕ̄_{y_{p}} belongs to a non-standard module W_{β} of dimension l and satisfying (2.49) (formally speaking, it holds modulo J but it is unimportant, see (3), proposition 3.3. Moreover, ϕ_{y_{p}} has form (2.64), hence ψ̄′_{y_{p}} ∈ W_{β} and therefore all operators Ē_{j}, j ∈ {1, …, l^{2}}, lie in A_{β}. From lemma 1.11 it follows that

\overline{ϕ_{y_{p}} ∗ tr(E)} = ϕ̄_{y_{p}} · tr_{A_{β}}(Ē) .

Next, consider the values of ϕ_{x}, where x ∈ Θ_{0}; by lemma 3.12, the representatives of all W_{β}, β ∈ pr(W), among x̄_{j}, occur “outside” ⋃_{x∈Θ_{0}} varb(ϕ_{x}), hence the values of all variables from varb(ϕ_{x}) are elements in simple components of D and in ⟨C⟩. Recall that the degree of ϕ_{x} is not less than the nilpotency index of ⟨C⟩, hence ϕ̄_{x} belongs to D. Also, the order of Θ_{0} is equal to dim(D), hence {ϕ̄_{x}| x ∈ Θ_{0}} is a basis of the vector space D. So, by lemma 1.11 we get

\overline{h ∗_{0} tr(E)} = h̄ · tr_{D}(Ē) .

Summarizing, we have

h̄′ = \overline{h ∗_{0} tr(E)} + ∑_{m,l≥1} \overline{h ∗_{m,l} tr(E)} = h̄ · tr_{U}(Ē) ,

so (3.31) holds in case h(x̄) ≠ 0; similarly, one can check the case h′(ȳ) ≠ 0. □

Equality (3.31) holds if we replace some variables with Lie polynomials; also it is linear with respect to h, hence (3.31) takes place for all h ∈ Υ. Clearly, it proves lemma 3.14.

Let m denote the maximum of dim(L_{i}), i ∈ {1, …, k}; take Y_{i} in the set of generic elements X_{i} of A with m variables and put Y = ⋃_{i=1}^{k} Y_{i}. Next, let B = (M; W) be the g.Lie submodule of F(A) generated by Y; also, put Υ̃′_{1} to be the set of all polynomials in Υ̃_{1} depending only on Y, i.e.,

Υ̃′_{1} = Υ̃_{1} ∩ W ,

and put B̂ = (M; Ŵ), where Ŵ = W/Υ̃′_{1}. Any polynomial in V ⟨X⟩, which is linear and skew-symmetric with respect to m + 1 variables in some X_{i}, is an identity of A. Hence, from proposition 1.1 it follows that T (B̂) = Υ_{1}, thus, it suffices to prove the representability of B̂, that is, the existence of a monomorphism from B̂ to a g.Lie module which is finite dimensional over a field extension of F.

Let Φ denote the subalgebra of all polynomials in K depending only on Y; recall that K is generated by tr_{U}(E), where E ∈ Π(U). Let Π(U)′ be the subalgebra of elements depending only on Y, then Φ is generated by tr_{U}(E), E ∈ Π(U)′. Moreover, from homogeneity of Π(L) and Φ it follows that Π(L)′ is integral over Φ.

Lemma 3.16. The algebra Φ is finitely generated over F.

Proof. Observe that Π(U)′ is a subalgebra of Pol(L⃗; End(U)), hence it is an associative PI algebra. Also, it is generated by a finite number of operators ad(x), x ∈ Y_{i}; besides, it is integral over Φ. Hence, by Shirshov’s theorem about height, e.g. [34, 37], Π(U)′ · Φ is a finitely generated module over Φ. Let E_{1}, …, E_{d}ıΠ(U)′ be the generators, then Φ is generated by the elements tr_{U}(E); hence Φ is a finitely generated algebra. □

Similarly, one can prove that Π(L)′ is a finitely generated module over Φ. Let N be the Φ-submodule generated by M in Pol(L⃗, L), i.e., N = Φ · M; observe that M is generated by Y as a module over Π(L)′, hence, N is a finitely generated Φ-module. Put N_{i} = Φ · M_{i}, then N = (N; N_{1}, …, N_{k}) is a k-sorted Lie algebra over Φ, which extends M.

Similarly, let U denote the extension of W by Φ, then C = (N; U) is a g.Lie module over Φ. Put Ĉ = (N; Û), where Û = U/Υ̃′_{1}, then B is embedded into C. Let P = En(Ĉ); since N is a finitely generated Lie algebra over Φ, P is also a finitely generated associative algebra . Besides, P inherits all polynomial identities of the finite dimensional algebra En(A), hence, it is PI. Finally, by lemma 2.14, P is a noetherian Φ-algebra, hence we can apply proposition 3.5 to the P -module Û.

Let H be the extension of the field F and let Û′ be a module over P′ = H ⊗_{F} P such that dim_{H}(Û′) < ∞ and Û′ extends Û as a module over P. Put N′ = H ⊗_{F} N and define a g.Lie representation ν : N′ → Û′ over H setting μ(α ⊗ a) = αμ(a) for all μ ∈ {ρ_{j}, δ_{j}}; this defines a g.Lie module C′ = (N′; Û′).

Clearly, we may assume that N′ (Û′) is the span of N (respectively, Û) over H. Hence, the ideal of identities of C′ (regarded as a g.Lie module over F) is equal to T (C) = Υ_{1}. Note that Ker(ν) is an ideal of finite codimension (cf. lemma 2.3), hence we can replace N′ with a finite dimensional k-sorted of Lie algebra over H. Proposition 3.4 is proven. □

So, let A′ = (L′; V′) be a finite dimensional g.Lie module over H whose ideal of identities (as a module over F) is equal to Υ_{1}.

Consider the variety of g.Lie modules over H of the same type as A′, denote it by 𝔐′_{k,s,r} to distinguish it from g.Lie modules over F. The free module M′⟨X⟩ is isomorphic to the scalar extension H ⊗_{F} M⟨X⟩, where we have the critical T-ideal Γ′ = H ⊗_{F} Γ with the same appropriate sequence of polynomials. Besides, A′ satisfies all identities of A, in particular, (2.19) and, therefore, L′ satisfies (2.20). Hence we have a “counter-example”

T (A′) ⊂ Γ′ .

Now, let us notice that the case ∥a(Γ, A)∥ = |b(Γ, A)| = 0 is impossible. Indeed, it means that D = 0 and W = 0 and therefore L = ⟨C⟩ is nilpotent. Consider the g.Lie submodule B = (M; W) of F(A) generated by the finite set Y = ⋃_{i=1}^{k} Y_{i}, where |Y_{i}| = dim(L_{i}). By proposition 1.1, chain of T-ideals (2.43) defines a strictly ascending chain of submodules of W over En(B). On the other hand, M is a finitely generated nilpotent Lie algebra, hence it is finite dimensional and, by lemma 2.14, W is a Noetherian module.

Fix a canonical decomposition of L′.

Lemma 3.17. The following inequalities hold:

∥a(Γ, A)∥ ≥ ∥a(Γ′, A′)∥  (3.32)
a(Γ, A) ≥ a(Γ′, A′) .  (3.33)

Moreover, if |b(Γ, A)| = 0, then ∥a(Γ, A)∥ > 0 and the inequality (3.33) is strict.

Proof. Put D′ = D(L′), W′ = W(L′) and so on. First, suppose that (3.32) is not true, then one can find x̄ ∈ V(Γ′, A′), such that ∥a(x̄)∥ > ∥a(Γ, A)∥; take a multilinear f ∈ Γ such that x̄ ∈ V(f, A′). To every α ∈ π(x̄) we fix a x_{α} ∈ varb(f) such that x̄_{α} ∈ D′_{α}; also, put n_{α} = dim_{H}(D′_{α}).

Let c ∈ ℕ such that ⟨C⟩^{c} = 0 and c ≥ than the nilpotency index of the radical of En(A). By lemma 2.11 and 2.10 there exist Lie polynomials ϕ_{α} over H and disjoint sets λ_{α}^{1}, …, λ_{α}^{c} of variables such that

(i) ϕ_{α} is a sum of multilinear polynomials; (ii) ϕ_{α} is multilinear and skew-symmetric with respect to variables of each λ_{α}^{j} and

|λ_{α}^{j}| = n_{α} for all j; (iii) ϕ_{α} is linear with respect to x_{α} and

x_{α} ∉ ⋃_{j=1}^{c} λ_{α}^{j} ;

also, ϕ_{α} ∈ L⟨X_{i}⟩ provided x_{α} ∈ X_{i}; (iv) under suitable substitution x → x̄ of elements from D′_{α} into ϕ_{α} we get ϕ̄_{α} = x̄_{α}. The point (iii) provides the following property. If we replace ȳ_{1}, …, ȳ_{q} in ϕ̄_{α}, where

y_{1}, …, y_{q} ∈ ⋃_{j=1}^{c} λ_{α}^{j}, by an element in ⋃_{γ≠α} D′_{γ}, then we get 0 in place of ϕ̄_{α}.

Further, let f_{1} denote the polynomial f where every x_{α} is replaced with ϕ_{α}; clearly, we have enough variables to assume that varb(ϕ_{α}) are disjoint sets, hence f_{1} is a sum of multilinear polynomials; besides f̄_{1} = f̄. For every j ∈ {1, …, c} put

Θ_{j} = ⋃_{α∈π(x̄)} λ_{α}^{j} ,

and let f_{2} denote the alternation of f_{1} with respect to the variables in Θ_{j} for all j; observe that f_{2} belongs to Γ′ thanks to (ii).

From (i)-(iv) it follows that f̄_{2} = q · f̄_{1} ≠ 0 for some q ∈ ℕ, hence f_{2} ∉ T (A′) = Γ′, hence f_{2} ∉ H ⊗_{F} T (A). Let g denote a homogeneous component of f_{2} which is not in T (A), hence g is multilinear and and skew-symmetric with respect to each Θ_{j}.

Moreover, since the c-th power of D_{i,α} is equal to itself, we can replace every element ȳ, y ∈ λ_{α}^{j} with a sum multilinear polynomials of degree ≥ c in suitable elements from D′_{i,α}. So we assume that every y ∈ λ_{α}^{j} is a multilinear polynomial of degree ≥ c.

On the other hand, H ⊗_{F} T (A) = T (H ⊗_{F} A); we claim that g is an identity of H ⊗_{F} A. Actually, suppose it does not vanish under a substitution x → x̃ of elements from H ⊗ L. Since g is multilinear, we may assume that all x̃ ∈ L. Moreover, f_{2} is a linear combination over H of polynomials from Γ, hence, so is g, hence x̃ ∈ Spl^{+}(L). Besides, every y ∈ Θ_{j} is a polynomial of degree ≥ c, hence we can assume that modulo J

ỹ ∈ ( ⋃_{α∈pr(D)} D_{α}) ∪ ( ⋃_{β∈pr(W)} W_{β}) .

Arguing in a standard way, one can show that for all j ∈ {1, …, c} there is x ∈ Θ_{j} such that x̃ belongs to W. Since c ≥ the nilpotency index of the radical of En(A), we get g̃ = 0. So, (3.32)) is proven.

In a similar fashion one may check (3.33)). First of all, recall that

a(Γ, A) = (a^{(1)}, …, a^{(m)}, …) ∈ ℕ_{0}^{∞} ,

where a^{(m)} is the number of α ∈ pr(D) such that the simple subalgebra D_{α} has dimension m; let D^{(m)} denote the sum of such subalgebras. Suppose that (3.33) is untrue, then there is x̄ ∈ V(Γ′, A′) such that a(Γ, A) < a(x̄), hence for some t ∈ ℕ we have a^{(t)} < a^{(t)}(x̄) and a^{(m)} = a^{(m)}(x̄) for all m > t.

Consider the polynomial g constructed above. Let Θ_{j}^{(m)} denote the union of all λ_{α}^{j} where the dimension of D′_{α} is equal to m, then

|Θ_{j}^{(t)}| > dim(D^{(t)})

and

|Θ_{j}^{(m)}| = dim(D^{(m)})

for all m > t. Hence,

∑_{m≥t} |Θ_{j}^{(m)}| > ∑_{m≥t} dim(D^{(m)}) ;  (3.34)

recall that every y ∈ Θ_{j}^{(m)} is supposed to be a multilinear polynomial of degree ≥ c; we can require an additional property: it is skew-symmetric with respect to m variables.

Now let us prove that g vanishes in H ⊗_{F} A; consider a substitution x → x̃. The polynomial g is multilinear, hence we may assume that x̃ ∈ Spl^{+}(L). By (3.34) for all j ∈ {1, …, c} there exists y in some Θ_{j}^{(m)}, m ≥ t, such that ỹ lies in W. Hence g̃ must be equal to 0, and (3.33) is also proven.

Finally, suppose |b(Γ, A)| = 0, then b(Γ, A) = (0, …, 0, …); hence, typical generators of (2.57) have a specific form: there are no sets Θ_{p} for all p ≥ 1. Suppose that (3.33) is an equality; put b = (0, …, 0, …) and c ∈ ℕ, arguing like in the preceding cases, one can construct a polynomial g ∈ S_{a,b,c}(H ⊗_{F} Γ^{#})) not vanishing under substitution of suitable elements in L′, which is impossible, because T (A′) contains all such g for sufficiently big c. Lemma 3.17 is proven. □

The type of T (A′) ⊂ Γ′ must be not less than type(Γ, A), hence, by lemma 3.17 a(Γ, A) = a(Γ′, A′); also, from corollary 3.2 we have t(Γ, A) = b(Γ, A). On the other hand,

S_{a,b,c}(H ⊗_{F} Γ) ⊆ H ⊗_{F} S_{a,b,c}(Γ) ⊆ T (A′) ,

hence t(Γ′, A′) < t(Γ, A) and the type of T (A′) ⊆ Γ′ is less than type(Γ, A), which is impossible. So, t(Γ, A) < b(Γ, A) and T-ideal (2.57), where b = b(Γ, A), is contained in T (A) for some c ∈ ℕ.

4. Canonical splitting

We are going to classify relations between simple components of D_{i} and that of W_{i} for each i; it turns out that there is a natural partition of these sets defined by action of L on W_{i}, which induces a decomposition of the algebra L_{i}/J_{i} into a direct sum of ideals. In fact, we split up ideals of a special form; the semisimple part is a simple algebra and the solvable radical is the direct sum of standard submodules of W_{i}.

Take α ∈ pr(D_{i}) and put

ω_{i,α} = {β ∈ pr(W_{i})| [W_{i,β}, D_{i,α}] ≠ 0} ;

in other words, ω_{i,α} consists of all projectors on simple components of W (not only W_{i}) with nontrivial action of D_{i,α}. The projector α is called isolated, if the following holds: (I1) α ∈ Ω = Ω(Γ, A), see (3.23); (I2) W_{i,β} is standard for all β ∈ ω_{i,α}.

If α is isolated, then for all α′ ∈ pr(D_{i}), α′ ≠ α, the subalgebra D_{i,α′} acts trivially on every W_{i,β}, β ∈ ω_{i,α}, hence

ω_{i,α} ∩ ω_{i,α′} = ∅ .  (3.35)

Also,

[W_{i,β}, C_{i}] = 0  (3.36)

modulo W_{i}^{[1]} ⊆ J_{i}.

Let π_{i}^{∘} denote the set of all isolated projectors in pr(D_{i}) and put π_{i}^{•} = pr(D_{i})\π_{i}^{∘}. To each α ∈ π_{i}^{∘} let L_{i}^{α} denote the ideal of L generated by

D_{i,α} ∪ ⋃_{β∈ω_{i,α}} W_{i,β} ;  (3.37)

also, put

ω_{i}^{∘} = ⋃_{α∈π_{i}^{∘}} ω_{i,α} ,  (3.38)

it is a disjoint union by (3.35); let

ω_{i}^{•} = pr(W_{i})\ω_{i}^{∘} ,  (3.39)

then for all α ∈ π_{i}^{∘} (α ∈ π_{i}^{•}) and β ∈ ω_{i}^{•} (β ∈ π_{i}^{∘}, respectively) we have

[W_{i,β}, D_{i,α}] = 0 .  (3.40)

Now, let L_{i}^{•} denote the ideal of L generated by

C_{i} ∪ ⋃_{α∈π_{i}^{•}} D_{i,α} ∪ ⋃_{β∈ω_{i}^{•}} W_{i,β} ,  (3.41)

then

L = ∑_{i=1} L_{i} = ∑_{i=1}^{k} ∑_{γ∈π_{i}^{∘}} L_{i}^{γ} ,

where π̃_{i}^{∘} = π_{i}^{∘} ∪ {•}. Put

k′ = ∑_{i=1}^{k} |π̃_{i}^{∘}| ,  (3.42)

then the formula above defines the structure of k′-sorted Lie algebra

L′ = (L; …, L_{i}^{γ}, …) .  (3.43)

Note that components of k′-sorting are enumerated by double indices, i ∈ {1, …, k} and γ ∈ π̃_{i}, so the standard notations used in the case of k-sorting are adopted in the same fashion.

The k′-sorted Lie algebra L′ has a canonical decomposition induced by that of L. First of all, by (2.20), (3.40) and (3.36) we have

I_{i}^{α}(L′) = ∑_{(j,γ)≠(i,α)}[L_{i}^{α}, L_{j}^{γ}] ⊆ N(L) ,

so L′ satisfies the analog of (2.20).

Suppose α ∈ π_{i}^{∘}, then D_{i,α} is the maximal semisimple subalgebra of L_{i}^{α}, so D_{i}^{α}(L′) = D_{i,α}; next, all simple components W_{i,β}, where β ∈ ω_{i,α}, are irreducible modules over D_{i,α}, hence

W = ∑_{β∈ω_{i,α}} W_{i,β}

lies in N_{i}^{α}(L′), hence S(L_{i}^{α}) = N_{i}^{α}(L′) and C_{i}^{α}(L′) = 0. Next,

M_{i}^{α}(L′) = N_{i}^{α}(L′)^{[1]} + N_{i}^{α}(L′)^{2} + I_{i}^{α}(L′) ⊆ M_{i} .

On the other hand, it contains W^{[1]} and the square of the algebra generated by W^{[0]}, hence, we can put W_{i}^{α} = W_{i}^{α}(L′) equal to W.

Next,

D_{i}^{•} = ⨁_{α∈π_{i}^{•}} D_{i,α}

is a maximal semisimple subalgebra of L_{i}^{•} and the radical S(L_{i}^{•}) is generated as an ideal of L by C_{i} and

W = ⨁_{β∈ω_{i}^{•}} W_{i,β} .

By lemma 2.5 and (3.35), all simple summands of W_{i}^{•} are nontrivial L_{i}^{α}-modules (with respect to the action induced by L), hence, N_{i}^{•}(L′) is the ideal generated by W and [C_{i}, C] + I_{i}^{•}; hence, we can put C_{i}^{•}(L′) = C_{i}. Finally, M_{i}^{•}(L′) ⊆ M_{i} contains [C_{i}, C] + I_{i}^{•} and W^{[1]}, hence, we set W_{i}^{•}(L′) = W_{i}^{•} equal to W.

In particular, we have the following decomposition

Gn_{i}(L) = ⨁_{γ∈π_{i}^{∘}} Gn_{i}^{γ}(L′) .  (3.44)

Now, put J_{i}^{γ} = J_{i} ∩ L_{i}^{γ} for all γ ∈ π̃_{i}; then J_{i}^{γ}(L′) ⊆ J_{i}^{γ} . From (3.40), (3.36) and (3.35) it follows that for all different α, β ∈ π̃_{i}

[L_{i}^{α}, L_{i}^{β}] ⊆ J_{i} ,  (3.45)

hence, the algebra L_{i}/J_{i} is decomposed into the sum of ideals L_{i}^{γ} + J_{i}/J_{i} ≃ L_{i}^{γ}/J_{i}^{γ}, we claim that it is direct.

First, observe that

L_{i}^{α}/J_{i}^{α}(L′) ≃ D_{i,α} + W_{i}^{α} = Gn_{i}^{α}(L′) ⊆ Gn_{i}(L)

for all α ∈ π_{i}^{∘}; since Gn_{i}(L) ∩ J_{i} = 0, we have

L_{i}^{α}/J_{i}^{α} ≃ Gn_{i}(L) ,

in particular, J_{i}^{α} = J_{i}^{α}(L′) in this case. Next,

L_{i}^{•}/J_{i}^{•} ≃ L_{i}^{•} + J_{i}/J_{i} ≃ (D_{i}^{•} ⊕ W_{i}^{•} ⊕ C_{i} ⊕ (⟨C_{i}⟩ + J_{i})) /J_{i} ≃
≃ Gn_{i}^{•}(L′) ⊕ (⟨C_{i}⟩^{2} + J_{i}/J_{i}) .

On the other hand,

L_{i}/J_{i} = Gn_{i}(L) ⊕ (M_{i}/J_{i}) ≃ Gn_{i}(L) ⊕ (⟨C_{i}⟩^{2} + J_{i}/J_{i})

which together with (3.44) gives the required decomposition,

L_{i}/J_{i} ≃ ⨁_{γ∈_{π_{i}}} L_{i}^{γ}/J_{i}^{γ} .  (3.46)

Let μ_{i}^{γ} denote the canonical epimorphism L_{i} → L_{i}/J_{i} → L_{i}^{γ}/J_{i}^{γ}; we need to extend it to a homomorphism L → L_{i}/J_{i}. Put J = ∑_{i=1}^{k} J_{i}, then

L/J = ∑_{i=1}(L_{i} + J)/J ;  (3.47)

if the sum is direct and J ∩ L_{i} = J_{i}, then we can extend μ_{i}^{γ} to L by replacing it with the composition of

(1) the canonical homomorphism L → L/J,

(2) the projection of L/J onto the direct summand L_{i} + J/J ≃ L_{i}/J_{i},

(3) μ_{i}^{γ}.

Since J_{i} ⊆ J ∩ L_{i}, we have a canonical homomorphism from L_{i}/J_{i} onto L_{i} + J/J. Put Gn_{i}^{+}(L) = D_{i} + W_{i} + ⟨C_{i}⟩, then L_{i} = Gn_{i}^{+}(L) + J_{i}, hence

dim(Gn_{i}^{+}(L)) ≥ dim(L_{i}/J_{i}) ≥ dim(L_{i} + J/J) .  (3.48)

Besides, modulo J the vector space L is equal to the sum

∑_{i=1}^{k} Gn_{i}^{+}(L) .  (3.49)

Suppose it is direct, then

∑_{i=1}^{k} dim (Gn_{i}^{+}(L)) = dim(L/J) ≤ ∑_{i=1}^{k} dim(L_{i} + J/J)

hence, in (3.48) we get equalities which proves the required decomposition of L.

To this end we are going to change slightly L. Let M = (M; M_{1}, …, M_{k}) denote the k-sorted subalgebra of L generated by ⋃_{i=1}^{k} D_{i} ∪ W_{i}^{[0]}; then M + ⟨C⟩ = L by lemma 2.7. Recall that M is an ideal of L, hence ⟨C⟩ acts by derivations on M.

Let q ∈ ℕ such that ⟨C⟩^{q} = 0; take a free product of Lie algebras ⟨C_{i}⟩, i ∈ {1, …, k}, in the variety of Lie algebras [5] and let N denote the quotient of this algebra over its q-th power. Then N is a finite dimensional nilpotent Lie algebra containing every ⟨C_{i}⟩, where the sum ∑_{i=1}^{k}⟨C_{i}⟩ is direct. Moreover, there exists a homomorphism π from N onto ⟨C⟩ restricting to the identity mapping on each ⟨C_{i}⟩. It induces a representation of N in Der(M), so M̃ = M ⊕ N is equipped with the structure of Lie algebra, so π̃ : a ⊕ b → a + π(b) is an algebra homomorphism of M̃ onto L.

Let N_{i} be the ideal of N generated by C_{i}, then M̃_{i} = M_{i}⊕N_{i} is an ideal M̃ and π̃(M̃_{i}) = L_{i}. Thus, π̃ is a homomorphism of the k-sorted Lie algebra M̃ = (M̃; M̃_{1}, …, M̃_{k}) onto L. Its composition with the g.Lie representation ν : L → End(V) is that of M̃; put Ã = (M̃; V), then T (A) = T (Ã).

We can put Gn_{i}(M̃) = Gn_{i}(L). Indeed, since the kernel Ker(π̃) coincides with Ker(π) ⊆ N, the subalgebra D_{i} ⊆ M̃_{i} is the maximal semisimple one and the radical S(M̃_{i}) is equal to the pre-image of S(L_{i}) = S(M_{i}) + ⟨C⟩ with respect to π̃, i.e., S(M_{i}) ⊕ N_{i}. Next, N_{i}(M̃) contains W_{i}^{[0]} ⊆ M_{i}, hence, N_{i}(M̃) = S(M_{i}) ⊕ [N_{i}, N] and we can put C_{i}(M̃) = C_{i} ⊆ N_{i}. Finally, W_{i} + M_{i}(M̃) is equal to N_{i}(M̃), hence we let W_{i}(M̃) = W_{i}.

Notice that the ”counter-example” T (A′) ⊂ Γ is irreducible with respect to this canonical decomposition. Indeed, π̃ maps Gn_{i}(M̃) onto Gn_{i}(L), hence, A′ is linearized and (3.2) takes place for all x̄ ∈ V(Γ, A′). Also, the image of J_{i}(A′) is equal to J_{i}, hence (3) of proposition 3.3 is valid; finally, the verbal ideal Γ^{#}[A′] is equal to Γ^{#}[A], hence it is an irreducible submodule of V lying in every nonzero one.

Summarizing, we get the following statement.

Lemma 3.18. There exists an irreducible ”counter-example” T (A) ⊂ Γ of minimal type such that J_{i}^{γ} = J_{i} ∩ L_{i}^{γ} is equal to J ∩ L_{i}^{γ} for all i, γ and L/J is decomposed into the direct sum of ideals L_{i}^{γ} + J/J,

L/J = ⨁_{i=1}^{k} ⨁_{γ∈π_{i}^{∘}} L_{i}^{γ}/J_{i}^{γ} .  (3.50)

So, hereafter we assume that T (A) ⊆ Γ is an irreducible “counter-example”, where L has a canonical decomposition satisfying (3.50). As a matter of fact, this property will be really working only in the last section.

5. Main Lemma

In this section we arrive to the point where the situation differs substantially from the associative case. In fact, Main Lemma describes in details the obstacle for generalization theorem 1.1 about representability of relatively free algebras to the class of Lie algebras. At the same time it helps us to find a ”counter-example” of lesser type in another class of g.Lie modules, so it will play the most important role in the rest.

Recall that t(Γ, A) < b(Γ, A), hence, by corollary 3.1,

S_{a,b,c}(Γ^{#})  (3.51)

where a = a(Γ, A) and b = b(Γ, A) , is contained in T (A) for suitable c ∈ ℕ.

Take x̄ ∈ V(f, A), where f ∈ Γ^{#}; from (3.2) and lemma 3.4 it follows that for each α ∈ pr(D_{i}) and β ∈ pr(W_{i}) there exist y_{α}, z_{β} ∈ varb(f) such that ȳ_{α} ∈ D_{i,α} and z̄_{β} ∈ W_{i,β}; moreover, since A is linearized, z_{β} is unique for each β. Put

Y_{i} = {y_{α}| α ∈ π_{i}^{∘}} ⊆ varb_{i}(f)  (3.52)

where every y_{α} is chosen in some way; also, let Y = ⋃_{i=1}^{k} Y_{i}. Next, put

Z_{i}^{α} = {z_{β}| β ∈ ω_{i,α}} , Z_{i} = ⋃_{α∈π^{∘}} Z_{i,α}  (3.53)

and Z = Z(f, x̄) = ⋃_{i=1}^{k} Z_{i}.

Lemma 3.19. (Main Lemma) For all multilinear f ∈ Γ^{#}, x̄ ∈ V(f, A) and Y, Z ⊆ varb(f) defined above, there exists a substitution x̃ ∈ V(f, A) such that: (M1) if x ∉ Y ∪ Z, then x̃ = x̄; (M2) for all i and α ∈ π_{i}^{∘}, if x ∈ {y_{α}} ∪ Z_{i,α}, then x̃ ∈ Spl_{i}^{α}(L′) ; (M3) there exists i and α′ ∈ π_{i}^{∘} such that ỹ_{α′} ∈ W_{i}^{α′}(L′).

Proof. Suppose, the lemma is untrue for some f, x̄ and Y. It means that for all substitutions x → x̃ ∈ Spl(L), satisfying (M1)-(M3) we have

f(x̃) = 0 .  (3.54)

Using this hypothesis, we will build an element from T-ideal (3.51), which is not an identity of T (A).

By definition, all nonstandard submodules of W_{i} are contained in W_{i}^{•}, let ω_{i}^{⋆} ⊆ ω_{i}^{•} denote the corresponding set of projectors. Also, for all α ∈ π_{i}^{•} ∩ Ω the set ω_{i,α} contains the projector on a nonstandard module (otherwise, α would be isolated); we fix one of them, say β; in this case we say that β is marked for α. For all β ∈ ω_{i}^{⋆} let m_{β} denote the set of all α for which β being marked; note that m_{β} can be empty.

Put q = |pr(W)|, by (3.2) it is equal to |b|. Take β ∈ ω_{i}^{⋆}, where dim(ad_{β}(L)) = m and m_{β} = {α_{1}, α_{2}, …, α_{n}}. From lemma 2.10, 2.11 and 2.12 it follows that there exits a Lie polynomial f_{β} ∈ L⟨X_{i}⟩ of the form

h (ϕ_{β}; ϕ_{α_{1}}^{(0)}, …, ϕ_{α_{1}}^{(q)}, …, ϕ_{α_{n}}^{(0)}, … ϕ_{α_{n}}^{(q)} …, ) ,  (3.55)

where (1) h is a Lie polynomial

h (y, x_{1}^{(0)}, …, x_{1}^{(q)}, …, x_{n}^{(0)}, …, x_{n}^{(q)} …, ) ∈ L⟨X_{i}⟩

which is linear with respect to y and all x_{j}^{(p)}, 1 ≤ j ≤ n, 0 ≤ p ≤ q;

(2) ϕ_{β}(z_{β}, …) is multilinear and skew-symmetric with respect to m + 1 variables including z_{β} (cf. lemma 2.12);

(3) each ϕ_{α_{j}}^{(p)} is linear and skew-symmetric with respect to dim(D_{i,α_{j}}) variables;

(2) there is a substitution x → x̄ such that for all x ∈ varb(ϕ_{α_{j}}^{(p)}) the element x̄ lies in D_{i,α_{j}} and ψ̄_{β} = z̄_{β} modulo J_{i}.

Note that ψ_{β} = ϕ_{β}, if m_{β} is empty; by lemma 2.5, [W_{i,β}, \underbrace{L_{i}, …, L_{i}}_{c}] = W_{i,β} modulo

J_{i}, hence, there exists a polynomial

g_{β} ∈ [f_{β}, \underbrace{L⟨X_{i}⟩, …, L⟨X_{i}⟩}_{c}]

and an extension of the substitution x → x̄ such that ḡ_{β} = z̄_{β}. Moreover, we can choose these polynomials depending on different variables and the substitution x → x̄ extends x̄, so f̄′ = f̄, where

f′ = f|_{z_{β}=g_{β} ∀β∈ω_{i}^{⋆} ∀i} .

We cannot require g_{β} to be multilinear, but we can assume that it is a sum of multilinear polynomials, so is f′; note that every homogeneous component of g_{β} is linear with rsepect to z_{β} and each λ_{α}^{(p)}.

Next, by definition, for each α ∈ π_{i}^{•}\Ω we have Ann(α) = 0, hence, there are ξ_{α} ∈ {ρ_{j}, δ_{j}} and x̄_{α} ∈ D_{i,α} such that

f̄· ∏_{i=1}^{k} ∏_{α∈π^{•}\Ω} ξ_{α}(x̄_{α}) ≠ 0 .

Like in the case above, for each α ∈ π_{i}^{•}\Ω we build a Lie polynomial

f_{α} = h (x_{α}, ϕ_{α}^{(0)}, …, ϕ_{α}^{(q)}, …) ,  (3.56)

where h = h(y, x^{(0)}, …, x^{(q)}, …) ∈ L⟨X_{i}⟩ is linear with respect to y and all x^{(p)}, each ϕ_{α}^{(p)} is linear and skew-symmetric with respect to dim(D_{i,α}) variables; and for a substitution of suitable elements in D_{i,α} we get ψ̄_{α} = x̄_{α}. Moreover, like in the case above we can find

g_{α} ∈ [f_{α}, \underbrace{L⟨X_{i}⟩, …, L⟨X_{i}⟩}_{c}] ,  (3.57)

which is a sum of multilinear polynomials, and an extension of the substitution x → x̄ such that ḡ_{α} = x̄_{α}. Put

f′′ = f′ · ∏_{i=1}^{k} ∏_{α∈π_{i}^{•}\Ω} ξ(g_{α}) ;

we assume that the substitution above extends x̄, hence f̄_{2} ≠ 0.

Finally, for all α ∈ π_{i}^{∘} we replace every y_{α} ∈ Y_{i} by a polynomial g_{α} of form (3.57) and extend the substitution x → x̄ so that ḡ_{α} = ȳ_{α} and the resulting polynomial, f′′′, does not vanish.

So, for all i, α ∈ pr(D) and p ∈ {0, 1, …, q} we have built a polynomial ϕ_{α}^{(p)} which is multilinear and skew-symmetric with respect to every set of variables λ_{α}^{p}, where |λ_{α}^{p}| = dim(D_{α}) for all p. Also, we can assume that for each α and p

(*) ϕ_{α}^{(p)} is linear with respect to some variable x ∉ ⋃_{p=0}^{q} λ_{α}^{(p)} .

Let pr(W) = {β_{1}, …, β_{q}}; for all β = β_{p} ∈ pr(W) put

Θ_{p} = {z_{β}} ∪ ⋃_{α∈pr(D)} λ_{α}^{(p)} ,

where z_{β} ∈ Z; in addition, let

Θ_{0} = ⋃_{α∈pr(D)} λ_{α}^{(0)} .

Notice that every homogeneous component of f′′′ has the form of f_{1} in Step 2 of the definition of (2.66); namely, the role of f is played by

f · ∏_{i=1}^{k} ∏_{α∈π^{•}\Ω} ξ_{α}(x_{α}),

where ϑ is equal to

Z ∪ {x_{α}| α ∈ π_{i}^{•}\Ω} ∪ {x_{α}| α ∈ π_{i}^{∘}}

and g_{x} = g_{β} (g_{x} = g_{α}) if x = z_{β} (x = x_{α}).

Let h′_{0} denote the alternation of f′′′ with respect to each Θ_{p}, 0 ≤ p ≤ q.

Lemma 3.20. Under the substitution x → x̄ the polynomial h′_{0} does not vanish.

Proof. Let G be the group ∏_{p=0}^{c} Sym(Θ_{p}), then

h′_{0} = ∑_{σ∈G} sgn(σ)σ · f′′′ .

Take σ ∈ G such that under x → x̄

\overline{σ · f′′′} ≠ 0 .  (3.58)

Let us show that σ(z_{β}) = z_{β} for all β ∈ pr(W).

First, suppose that W_{i,β} is non-standard, i.e., β ∈ ω_{i}^{⋆}. If σ(z_{β}) ≠ z_{β} then y = σ^{−1}(z_{β}) lies in some λ_{j}^{(p)}, hence for all x ∈ varb(σ ·ϕ_{β}) the element x̄ belongs to D∪C. Recall that ϕ_{β} is linear and skew-symmetric with respect to m + 1-variables, z_{β} and, say, x_{1}, …, x_{m}, hence, σ · ϕ_{β} has the same property for y, x_{1}, …, x_{m}.

By definition, x̄ ∈ D ∪ C for all x ∈ varb(ϕ_{β}), x ≠ z_{β}; hence x̄ ∈ D ∪ C for all x ∈ varb(σ · ϕ_{β}). Suppose, there is a variable among y, x_{1}, …, x_{m} which belongs to C, then all variables of σ·ϕ_{β} must take value in C too. Recall that degree of each homogeneous component of ϕ_{β} is not less than nilpotency index of C, hence \overline{σ · ϕ_{β}} must be equal to 0 and (3.58) does not hold. So, x̄_{1}, …, x̄_{m} lies in some D_{i,α} which is isomorphic to a subalgebra of ad_{β}(L), hence ȳ ∈ D_{i,α} too. Besides, dim(D_{i,α}) ≤ dim(ad_{β}(L)) = m, hence, \overline{σ · ϕ_{β}} = 0 modulo J_{i}, which is impossible. Thus, σ must fix z_{β}.

Now suppose that W_{i,β} is standard, then there exists only one α ∈ pr(D_{i}) such that [W_{i,β}, D_{i,α}] ≠ 0, hence, by (*), σ(z_{β}) ∈ λ_{α}^{(p)}. We have three cases: α ∈ π_{i}^{•} ∩ Ω, α ∈ π_{i}^{•}\Ω and α ∈ π_{i}^{∘}. In the first case, α ∈ m_{β′} for some β′ ∈ ω_{j}^{⋆}; hence, σ(z_{β}) ∈ varb(f_{β′}). By the previous case, σ(z_{β′}) = z_{β′}, hence, \overline{σ · f_{β′}} belongs to the ideal J_{i}, and, by lemma 3.3, inequality (3.58) is impossible. Next, if α ∈ π_{j}^{•}\Ω, then \overline{σ · ϕ_{α}^{(p)}} lies in the ideal generated by W_{i}, hence, by lemma 3.9, (3.58) does not hold. Finally, if α ∈ π_{j}^{∘}, then σ(z_{β}) ∈ Y. Hence,

\overline{σ · f′′′} = f(x̃_{1}, …, x̃_{n}) · ∏_{α∈π^{•}\Ω} ξ_{α}(x̄_{α}) ,

where x → x̃ is the substitution satisfying (M1)-(M3), hence, \overline{σ · f′′′} must equal 0 which contradicts to (3.58).

So, σ(z_{β}) = z_{β} for all β; suppose β = β_{p}, then for all α and x ∈ varb(σ · ϕ_{α}^{(p)}) we have x̄ ∈ D_{i} ∪ C_{i}, hence , by (*), σ(λ_{α}^{(p)}) = λ_{α}^{(p)} for all α. Similarly, if α ∈ pr(D_{i}), then σ(λ_{α}^{(0)}) = λ_{α}^{(0)}. Thus, if \overline{σ · f′′′} ≠ 0 , then σ belongs to the direct product of groups,

G_{1} = ∏_{p=0}^{q} ∏_{α∈pr(D)} Sym(λ_{α}^{(p)}) .

Therefore, h̄′_{0} = |G_{1}| · f̄′′′ ≠ 0, so lemma 3.20 is proven. □

To obtain an element in S_{a,b,c}(Γ^{#})\T (A) we make the following substitution into h′_{0}: every x ∈ λ_{α}^{(p)}, p ∈ {0, …, q}, is replaced with a Lie polynomial ϕ_{x} defined as follows. By lemmas 2.10 and 2.11, there exists a polynomial ϕ_{x} ∈ LCap_{m,1}^{2}, where m = dim(D_{i,α}), with homogeneous components of degree ≥ d_{i} such that ϕ̄ = x̄ under some substitution of elements in D_{i,α}. Clearly, it belongs to the T-ideal generated by all Lie polynomials of form (2.63).

Next, every z_{β} ∈ Θ_{p}, where β = β_{p}, plays the role of the fixed element y_{p}; by lemma 2.12, there exists a polynomial ϕ_{y_{p}} from the T-ideal generated by (2.64), where m = dim(ad_{β}(L)) and l = dim(W_{β}) such that ϕ̄_{y_{p}} = z̄_{β} under a suitable substitution. Thus, replacing every x ∈ Λ_{p} with ϕ_{x} in h′_{0} we get a polynomial h′ from S_{a,b,c}(Γ^{#}) which does not vanish under the substitution x → x̄. □

6. Sorting of variables

Put Υ = Γ^{#} and take a multilinear f = f(x_{1}, …, x_{n}) ∈ Γ^{#}; modulo T (A) it can be regarded as a function in Pol(L⃗; V) in generic elements x_{1}, …, x_{n}. Besides, from (3.46) it follows that modulo J_{i} every x_{j} is decomposed into the sum of μ_{i}^{γ}(x_{j}), where γ ranges over π̃_{i}. Hence, by (3) of proposition 3.3, the function f is represented as

f(∑_{μ_{1}} μ_{1}(x_{1}), …, ∑_{μ_{n}} μ_{n}(x_{n})) = ∑_{μ_{1},... ,μ_{n}} f(μ_{1}(x_{1}), …, μ_{n}(x_{n})) ,  (3.59)

where μ_{j} runs over {μ_{i}^{γ}, γ ∈ π̃_{i}^{∘}} provided x_{j} ∈ X_{i}; note that

g(x_{1}, …, x_{n}) = f(μ_{1}(x_{1}), …, μ_{n}(x_{n}))  (3.60)

is a multilinear mapping in Pol(L⃗, V).

Let F(L) be the k-sorted Lie algebra of generic elements of L, see (2.18); it is contained in Pol(L⃗; L), hence the g.Lie representation ν : L → End(V) induces that of F(L) in End (Pol(L⃗, V)). A subspace of Pol(L⃗, V) is said to be fully characteristic, if it is stable under substitutions of F(L)_{i} in place of generic elements in X_{i}. Let Υ̃ denote the fully characteristic subspace of Pol(L⃗, V) generated by all elements (3.60).

Proposition 3.6. Υ̃ is a finitely generated fully characteristic submodule over En(F(L), ν).

The proposition states the following: there exist functions g_{1}, …, g_{t} of form (3.60) such that Υ̃ is equal to the span of elements

g_{p}(u_{1}, …, u_{n}) · E ,  (3.61)

where p ∈ {1, …, t}, u_{j} ∈ F(L)_{i} provided x_{j} ∈ X_{i} and E ∈ En(F(L), ν).

We will rewrite this statement in terms of identities of g.Lie modules in the variety 𝔑 = 𝔐_{k′,s,r}, where k′ is given by (3.42); then, using Main Lemma, we will pass to another variety with a “counter-example” of type < type(Γ, A).

Consider a k′-sorted set of free variables :

X′ = ⋃_{i=1}^{k} ⋃_{γ∈π^{∘}} X_{i}^{γ}  (3.62)

where X_{i}^{γ} = {x^{γ}| x ∈ X_{i}}; it can be considered as a k-sorted set of free variables with

X′_{i} = ⋃_{γ∈π^{∘}} X_{i}^{γ} .

Let M⟨X′⟩ be a free g.Lie module in 𝔐 with the set of free generators X′. If I is a T-ideal of M⟨X⟩, then denote by I′ the verbal ideal in M⟨X′⟩; as a vector space, I is generated by elements of the form

f(x_{1}^{γ_{1}}, …, x_{n}^{γ_{n}}) ,  (3.63)

where f ∈ I and x_{j}^{γ} ∈ X′_{i} provided x_{j} ∈ X_{i} (in other words, we replace the set of variables X_{i} with X′_{i}). In particular, for all T-ideals I, J of M⟨X⟩ we have

I ⊈ J ⟺ I′ ⊈ J′ .  (3.64)

Put

B = (L′; V′; ν, v_{1}, …, v_{r}) ∈ 𝔑 ,  (3.65)

where L′ is the k-sorted Lie algebra (3.43); it is a finite dimensional g.Lie module in 𝔑. Similarly, define a k′-sorting of the Lie algebra L⟨X′⟩ setting the (i, γ)-component to be the ideal L_{i}^{γ}⟨X⟩ generated by X_{i}^{γ}, so

L′⟨X′⟩ = (L⟨X′⟩; …, L_{i}^{γ}⟨X⟩, …) .

Observe that U(L⟨X′⟩, ν) = U(L⟨X′⟩, ν) = U(L′⟨X′⟩, ν), hence,

N ⟨X′⟩ = (L′⟨X′⟩; V ⟨X′⟩) ,  (3.66)

where V ⟨X′⟩ is the second coordinate of M⟨X′⟩, is the free g.Lie module in 𝔑. Note that every T-ideal I ⊆ V ⟨X′⟩ of M⟨X′⟩ is that of N ⟨X′⟩. Put Υ = Γ^{#}.

Proposition 3.7. The T-ideal Υ′ is finitely generated modulo T (B).

First of all, we show that it yields proposition 3.6. Define a mapping

θ : Υ̃ → Υ′/T (B) ;

take g ∈ Υ̃; it can be represented in the form

g = ∑_{f;μ_{1},... ,μ_{n}} f(μ_{1}(x_{1}), …, μ_{n}(x_{n})) ,  (3.67)

where f ∈ Υ and μ_{1}, …, μ_{n} ∈ {μ_{i}^{γ}| i, γ}. Put

θ(g) = ∑_{f;μ_{1},... ,μ_{n}} f(x_{1}^{γ_{1}}, …, x_{n}^{γ_{n}}) ∈ Υ′/T (B) ,  (3.68)

where γ_{j} is defined by equality μ_{j} = μ_{i}^{γ_{j}}. Observe that the definition is correct; for it suffices to check that g = 0 implies θ(g) = 0. On the contrary, suppose that θ(g) do not vanish under x → x̄; by defintion, x̄_{j}^{γ_{j}} must be in L_{i}^{γ_{j}}, where γ_{j} ∈ π̃_{i}^{∘}. On the othe hand, the projector γ_{j} has been defined by the equality μ_{j} = μ_{i}^{γ_{j}}. Hence, μ_{j}(x̄_{j}^{γ_{j}}) = x̄_{j}^{γ_{j}} modulo J_{i} and

∑_{f;μ_{1},... ,μ_{n}} f(μ(x̄_{1}), …, μ(x̄_{n})) = \overline{θ(g)} ≠ 0 ,

which contradicts to the hypothesis.

Lemma 3.21. θ is bijective.

Proof. Clearly, θ is onto; let us show that Ker(θ) = 0. Take g ∈ Υ̃ of form (3.67) and suppose that g does no vanish under x → x̄. If x_{j} ∈ X_{i}, then x̄_{j} ∈ L_{i}; take x̄_{j}^{γ_{j}} ∈ L_{i}^{γ_{j}} such that μ_{j}(x̄_{j}) = x̄_{j}^{γ_{j}} modulo J_{i}. By irreducibility of the ”counter-example” (see (3) of proposition 3.3)

∑_{f;μ_{1},... ,μ_{n}} f(x̄_{1}^{γ_{1}}, …, x̄_{n}^{γ_{n}}) = ∑_{f;μ_{1},... ,μ_{n}} f(μ(x̄_{1}), …, μ(x̄_{n}))

hence θ(g) takes a nonzero value under the substitution x_{j}^{γ_{j}} → x̄_{j}^{γ_{j}}. □

Let h_{1}, …, h_{t} be multilinear generators of Υ′ modulo T (B), we can assume that they are of form (3.63), where f_{l} = f_{l}(x_{1}, …, x_{n}) ∈ Υ, l = 1, …, t. Hence, Υ′ is spanned modulo T (B) by polynomials

f_{l}(u_{1}, …, u_{n}) · E  (3.69)

where u_{j} ∈ L_{i}^{γ}⟨X′⟩ provided μ_{j} = μ_{i}^{γ}, and E is a monomial in ν(X′).

To prove proposition 3.6 it suffices to show that the pre-image of (3.69) belongs to the fully characteristic submodule generated by g_{l} = f(μ_{1}(x_{1}), …, μ_{n}(x_{n})) = θ^{−1}(h_{l}), l ∈ {1, …, t}. Let g be the polynomial obtained from (3.69) by replacing every variable x^{γ} ∈ X_{i}^{γ} with μ_{i}^{γ}(x),

g = f(u′_{1}, …, u′_{n})E′,

then θ(g) is equal to (3.69) by definition. Recall that μ_{j} is a homomorphism from L_{i} to L_{i}^{γ_{j}}/J_{i}^{γ_{j}}, hence, we can replace u′_{j} with μ_{j}(v_{j}), where v_{j} ∈ L_{i}⟨X⟩. Also, as π_{i}^{∘} ⊆ Ω(Γ, A), in E′ every operator δ_{p}(μ_{j}(x)) (or ρ_{p}(μ_{j}(x))) can be replaced with δ_{p}(x) (ρ_{p}(x), respectively) for all x ∈ X. Therefore, the pre-image of (3.69) belongs to the fully characteristic submodule generated g_{1}, …, g_{t}, which gives proposition 3.6.

Proof of proposition 3.7. Suppose it is untrue, then there exists a critical T-ideal Ψ ⊆ Υ′ + T (B), where T (B) ⊂ Ψ and Ψ^{#} ⊆ Υ′; by (3.45), we have Υ′ ⊆ Q(B), hence, T (B) ⊂ Υ′ is a ”counter-example” in 𝔑; observe that it inherits properties (1)-(3) in proposition 3.3. Let us show that its type is ≤ type(Γ, A).

Recall that

Spl(L′) = ⋃_{i=1}^{k} ⋃_{α∈π_{i}^{∘}} Spl_{i}^{α}(L′) = ⋃_{i=1}^{k} Spl_{i}(L) = Spl(L) ,

also, (3.2) holds for all x̄, hence

∥a(Ψ, B)∥ ≤ ∥a(Γ, A)∥ , a(Ψ, B) ≤ a(Γ, A),  (3.70)

|b(Ψ, B)| ≤ |b(Γ, A)| , b(Ψ, B) ≤ b(Γ, A) ,  (3.71)
ht(Ψ, B) ≤ ht(Γ, A) .  (3.72)

Since the type of the ”counter-example” T (A) ⊂ Γ is the smallest one, (3.70) must be equalities. Now we need to show that

t(Ψ, B) ≤ t(Γ, A) .  (3.73)

By the definition of (2.57), for all b ∈ (ℕ_{0}^{∞})^{∞} and c ∈ ℕ

S_{a,b,c}(Ψ) ⊆ S_{a,b,c}(Υ′) ⊆ (S_{a,b,c}(Υ))′ ,

hence, if S_{a,b,c}(Ψ) ⊈ T (B), then (S_{a,b,c}(Υ))′ ⊈ T (A)′, because T (A)′ ⊆ T (B); by (3.64)

S_{a,b,c}(Υ) ⊈ T (A)

which proves (3.73).

To simplify notations, we put k = k′, hence 𝔐 becomes 𝔑. Also, now Γ means Ψ and A denotes B, where L_{1}, …, L_{m}, m ≤ k, corresponds to L_{i}^{γ} with γ running over π_{i}^{∘} and i ∈ {1, …, k}. Therefore, the other components, L_{m+1}, …, L_{k}, are all L_{i}^{•}. So, for all i ≤ m the algebra D_{i} is simple, C_{i} = 0 and all W_{i,β} are standard.

Under this change of indices Main Lemma takes the following form. Let x̄ ∈ V(f, A), where f is a multilinear polynomial in Γ^{#}; let Y = {y_{1}, …, y_{m}} ⊆ varb(f) such that ȳ_{i} ∈ D_{i} for all i ≤ m; also, put Z_{i} = {z_{β}| β ∈ ω_{i}} and Z = ⋃_{i=1}^{k} Z_{i}.

Lemma 3.22. For any multilinear f ∈ Γ^{#}, x̄ ∈ V(f, A) and Y ⊆ varb(f) defined above, there exists a substitution x̃ ∈ V(f, A) such that: (M1’) if x_{j} ∉ Y ∪ Z, then x̃_{j} = x̄_{j}; (M2’) for all i ≤ m if x_{j} ∈ {y_{i}} ∪ Z_{i}, then x̃_{j} ∈ Spl_{i}(L) ; (M3’) ỹ_{i} ∈ W_{i} for some i ≤ m.

Proof. To check it we have to use the ”old” notations. Recall that Γ means Ψ, since f ∈ Ψ^{#} ⊆ Υ′, it has the form f′(x_{1}^{γ_{1}}, …, x_{n}^{γ_{n}}) for some multilinear f′ ∈ Υ (so, x_{1}, …, x_{n} are different). Next, every y_{α} ∈ Y, where α ∈ π_{i}^{∘}, belongs to X_{i}^{α}, i.e., y_{α} = x_{j}^{γ_{j}} for some j (where γ_{j} = α); put y′_{α} = x_{j} ∈ X_{i} and let Y′_{i} = {y′_{α}|α ∈ π_{i}^{∘}} and Y′ = ⋃_{i=1}^{k} Y′_{i} ⊆ varb(f′). Observe that f′, Y′ and the substitution x_{j} → x̄_{j}^{γ_{j}} satisfy the hypothesis of Main Lemma; so there exists x → x̃ with properties (M1)-(M3). Hence, the substitution x̃_{j}^{γ_{j}} = x̃_{j} satisfies (M1’)-(M3’) (translated into ”old” notations). □

Now we are going to define a mapping B → B^{∼} from 𝔐 to a variety 𝔐_{k,s^{∼},r^{∼}}, which gives a ”counter-example” of lesser type; in fact, it will be similar to examples given in section 3.

Fix y_{i} ∈ X_{i} for all i ∈ {1, …, m}, put Y = {y_{1}, …, y_{m}}, X′_{i} = X_{i}\{y_{i}} and X′ = ⋃_{i=1}^{k} X′_{i}. Let V ⟨Y; X′⟩ denote the subspace of all polynomials from V ⟨X⟩ which are linear with respect to each y_{i} ∈ Y. As a T-ideal Γ is generated by its multilinear elements; moreover, every multilinear f ∈ Γ^{#} of degree 0 with respect some X_{i}, i ≤ m, must be an identity of A thanks to irreducibillity of the ”counter-example”. Hence, modulo T (A) the T-ideal Γ is generated by Γ ∩ V ⟨Y; X′⟩ and if f_{n}, n ∈ ℕ, is the appropriate sequence for Γ, then we may assume that each f_{n} depends on all y_{i} ∈ Y, i.e. f_{n} ∈ V ⟨Y; X′⟩.

Lemma 3.23. V ⟨Y; X′⟩ is spanned by elements of the form

h(u_{1}, …, u_{m}) · E

where u_{i} belongs to the submodule of L⟨X⟩ over L⟨X′⟩ generated by y_{i}, E is a product of operators in ν(X′) and h runs over the set

{v_{j}ξ_{1}(y_{i_{1}}) … ξ_{m}(y_{i_{m}})| ξ_{l} ∈ ⋃_{p=1}^{s}{ρ_{p}, δ_{p}} ,  (3.74)

j ∈ {1, …, r}, {i_{1}, …, i_{m}} = {1, …, m}} .

Proof. Indeed, V ⟨Y; X′⟩ is spanned by elements

v_{j} · w_{1} … w_{m}E

where w_{p} lies in the submodule over ν(M⟨X′⟩) ⊆ ν(M⟨X⟩) generated by ξ(y_{i_{p}}), ξ ∈ ⋃_{p=1}^{s}{ρ_{p}, δ_{p}}, and E is a product of operators in ν(X′). Using (2.10)-(2.12), we prove the statement. □

Let h_{1}, …, h_{r^{∼}} be all elements in (3.74). Take B = (M; W) ∈ 𝔐 and define B^{∼} = (M; W^{∼}) ∈ 𝔐^{∼} = 𝔐_{k,s^{∼},r^{∼}}, where s^{∼} = s + m. Let U = U(B) denote the vector space of all multilinear mappings from M_{1} × … × M_{m} to V; every h_{j} defines a function in U:

(a_{1}, …, a_{m}) → h_{j}(a_{1}, …, a_{m}) .

For all i ∈ {1, …, m} and j ∈ {1, …, k} define ρ_{i}^{∼} : M → End(U) setting

h · ρ_{i}^{∼}(a) = h|_{y_{i}=[a,y_{i}]} ;

also, we put δ_{i}^{∼} = 0. Furthermore, for all i ∈ {1, …, s}, we put ρ_{i+m}^{∼}(x) to be the operator R_{ρ_{i}(x)} : h → h · ρ_{i}(x); similarly, δ_{i+m}^{∼}(x) = R_{δ_{i}(x)}. Finally, let

ν^{∼} = (ν_{1}^{∼}, …, ν_{s^{∼}}^{∼})

where ν_{i} = (ρ_{i}^{∼}, δ_{i}^{∼}); one may check that all identities (2.10)-(2.11) take place, hence ν^{∼} is a g.Lie representation of the k-sorted Lie algebra M in End(U). Let W^{∼} denote the submodule over En(L, ν^{∼}) generated by v_{1}^{∼}, …, v_{r^{∼}}^{∼}. Then

B^{∼} = (M; W^{∼}; ν^{∼}; h_{1}, …, h_{r^{∼}})

is a g.Lie module in 𝔐^{∼}.

Let M be the free g.Lie module M⟨X′⟩ in the variety 𝔐 with the set of free variables X′ = X\Y. We identify V ⟨Y; X′⟩ with multilinear mappings in U(M); by definition, V ⟨X′⟩^{∼} ⊆ V ⟨Y, X⟩, hence, by lemma 3.23,

V ⟨Y; X′⟩ = V ⟨X′⟩^{∼}  (3.75)

Next, let N ⟨X′⟩ = (L⟨X′⟩; W ⟨X′⟩) denote the free g.Lie module of 𝔐^{∼}; the identity mapping of L⟨X′⟩ extends to an epimorphism (id, τ) from N onto M^{∼}.

Lemma 3.24. Every homomorphism from N to B^{∼} can be represented as the composition of (id, τ) and a suitable homomorphism from M^{∼} to B^{∼}.

Proof. Indeed, let (ϕ, ψ) : N ⟨X′⟩ → B^{∼} be a homomorphism, then ϕ : L⟨X′⟩ → M is a homomorphism of k-sorted Lie algebras. Define a mapping ζ : V ⟨Y; X′⟩ → U(B); take f = f(y_{1}, …, y_{m}) ∈ V ⟨Y; X′⟩, then ζ(f) is the following mapping in U(B). Take ȳ_{i} ∈ M_{i} for i ≤ m and extend ϕ to the homomorphism ϕ′ : L⟨X⟩ → M, letting ϕ′(y_{i}) = ȳ_{i}; it induces a homomorphism (ϕ′, ψ′) from M⟨X⟩ to B. Put ζ(f)(ȳ_{1}, …, ȳ_{m}) = ψ′(f); in other words, ζ(f) is obtained from the polynomial f = f(y_{1}, …, y_{m}, x_{1}, …, x_{n}) by substitution of ϕ(x_{j}) in place of x_{j}.

Let us check that ψ = ζ ∘ τ; i.e, for all f ∈ W ⟨X′⟩ we have ψ(f) = ζ(τ(f)). Recall that f = ∑_{j=1}^{r^{∼}} v_{j}^{∼} · E_{j}, so we can use the induction on the operator length of E_{j}. First, if f = v_{j}^{∼}, then ψ(f) = h_{j} = ζ(τ(f)); next, let ξ ∈ {ρ_{i}^{∼}, δ_{i}^{∼}} and x ∈ X′, we need to show that

ψ(f · ξ(x)) = ζ(τ(f · ξ(x))) .

Since (ϕ, ψ) is a homomorphism, the left hand side is equal to

ψ(f) · ξ(ϕ(x)) = ζ(τ(f)) · ξ(ϕ(x)) ;

we need to show that

ζ(τ(f)) · ξ(ϕ(x))(ȳ_{1}, …, ȳ_{m}) = ζ(τ(f · ξ(x)))(ȳ_{1}, …, ȳ_{m})

for all ȳ_{i} ∈ M_{i}. For example, let ξ = ρ_{1}, then, by the definition of ζ the left hand side is equal to

ζ(τ(f))([ϕ(x), ȳ_{1}], …, ȳ_{m}) =
= ψ′(τ(f))([ϕ(x), ȳ_{1}], …, ȳ_{m}) = ψ′(τ(f))([ϕ′(x), ϕ′(y_{1})], …, ϕ′(y_{m})) ;

further, since (ϕ′, ψ′) is a homomorphism, the latter term can be rewritten as

ψ′(τ(f)([x, y_{1}], …, y_{m}) = ψ′(τ(f) · ξ(x)) = ζ(τ(f) · ξ(x))(ȳ_{1}, …, ȳ_{m}) .

One can check similarly the other cases of ξ. □.

From the lemma it follows that the full pre-image of T (B) ∩ V ⟨Y; X′⟩ with respect to τ coincides with T (B^{∼}) for all B ∈ 𝔐. Indeed, if τ(f) ∈ T (B)∩V ⟨Y; X′⟩, then f vanishes under any homomorphism from N ⟨X′⟩ to B.; the reciprocal inclusion is evident.

Since every T-ideal is equal to T (B) for suitable B, the full pre-image τ^{−1}(I∩V ⟨Y; X′⟩), is a T-ideal. Put Υ = τ^{−1}(Γ ∩ V ⟨Y; X′⟩) and take g_{n} ∈ τ^{−1}(f_{n}), then g_{n}, n ∈ ℕ, is an appropriate sequence for Υ (observe that τ preserves the degree, because it is induced by the identity mapping of L⟨X′⟩). Besides, Υ ⊆ Q(A^{∼}), hence T (A^{∼}) ⊂ Υ is a ”counterexample” in 𝔐^{∼}. Moreover, from definition it follows that type(Υ, A^{∼}) ≤ type(Γ, A).

Actually, since the first component of A^{∼} is L, we have the analog of inequalities (3.70)-(3.72) in this case. Thus, we may assume that

a = a(Γ, A) = a(Υ, A^{∼})

and we only need to show the following: if

S_{a,b,c}(Υ) ⊈ T (A^{∼})  (3.76)

for some b and c, then S_{a,b,c}(Γ) ⊈ T (A). Take a typical generator (2.66) of the left hand side of (3.76) and suppose h ∉ T (A^{∼}), then τ(h) ≠ 0. On the other hand, τ(h) = τ(S_{a,b,c}(f)) = S_{a,b,c}(τ(f)); since τ(f) ∈ Υ = Γ^{#}, we get the required relation.

Take β ∈ pr(W), hence β ∈ pr(W_{l}) for some l; consider the k-sorted Lie subalgebra M^{β} of L generated by (3.13) and Gn_{i}(L), i ≠ l; it generates a g.Lie submodule C^{β} of A^{∼}.

Lemma 3.25.

T (A^{∼}) = Υ ∩ ⋂_{βpr(W)} T (C^{β}) .

Proof. Let Υ′ denote the right hand side, then T (A^{∼}) ⊆ Υ′, and we need the reciprocal inclusion. Suppose it is untrue; take a multilinear g ∈ Υ′\T (A^{∼}), then there is x̄ = (x̄_{1}, …, x̄_{n}) ∈ V(g, A^{∼}). Put f = τ(g) ∈ Γ ∩ V ⟨Y; X′⟩, then f = f(y) = (y_{1}, …, y_{m}, x_{1}, …, x_{n}) and

g = g(x̄) = f(y_{1}, …, y_{m}, x̄_{1}, …, x̄_{n})

is a nonzero mapping in V^{∼}. Hence, there exists ȳ_{i} ∈ L_{i} such that g(ȳ) ≠ 0; by irreducibilty of the ”counter-example” we may assume that each ȳ_{i} lies in Spl_{i}(L). By lemma 3.22, there exists ỹ ∈ V(f, A) such that some ỹ_{i} lies in W_{i,β} for suitable β. Recall that A^{∼} is linearized, hence, all x̃_{j} belong to M^{β}, which is impossible, because f ∈ T (C^{β}). □

By lemma 3.7

type(Υ, C^{β}) < type(Υ, A^{∼}) = type(Γ, A) ,

so, proposition 3.7 follows from lemma 2.15.

7. Final step

Let g_{1}, …, g_{t} be the functions from proposition 3.6,

g_{l} = f_{l}(μ_{1}(x_{1}), …, μ_{n}(x_{n})) ,  (3.77)

where the variables x_{j} and mappings μ_{j} depend on l as well as the number n, i.e., x_{j} = x_{j}^{(l)}, μ_{j} = μ_{j}^{(l)} and n = n^{(l)}. Let q denote the maximum of n^{(l)}, l ∈ {1, …, t}. We are going to define a g.Lie module A^{⋄} in the variety 𝔐^{⋄} = 𝔐_{k,s^{⋄},t}, where s^{⋄} = q + s.

Let y_{i} denote the projector of qL onto the i-th summand; consider the vector space U of all mappings from the direct sum qL to V that are linear or constant with respect to every y_{i}. For all i ∈ {1, …, q} define linear mappings ρ_{i}^{⋄}, δ_{i}^{⋄} : L → End(U) by formulas (2.8), (2.9)); also, for all i ∈ {1, …, s} and a ∈ L put ρ_{q+i}^{⋄}(a) = R_{ρ_{i}(a)}, δ_{q+i}^{⋄}(a) = R_{δ_{i}(a)}. One may check that identities (2.10)-(2.11) hold, hence,

ν^{⋄} = (ν_{1}^{⋄}, …, ν_{s^{⋄}}^{⋄})

is a g.Lie representation of the the Lie algebra L; it induces a g.Lie representation of the k-sorted Lie algebra L. Take l ∈ {1, …, t} and consider (3.77); recall that every μ_{j} = μ_{i}^{γ} extends to a homomorphism L → L_{i}^{γ}/J_{i}^{γ}, see (3.50); hence (3.77) defines a function h_{l} ∈ U.

Let V^{⋄} denote the submodule over En(L, ν^{⋄}) generated by h_{1}, …, h_{t} and put

A^{⋄} = (L; V^{⋄}; ν^{⋄}; h_{1}, …, h_{t}) ,

it is a f.d. generalized module in 𝔐^{⋄}.

Consider the g.Lie module of generic elements of A, see (2.18); let U be the vector space of mappings from q · F(L) to Pol(L⃗, V), which are linear or constant with respect to every generic element of M. For example, every h_{l} defines a mapping in U:

h_{l} : (u_{1}, …, u_{n}) → f_{l}(μ_{1}(u_{1}), …, μ_{n}(u_{n})) .

Like in case of A^{⋄} we define a g.Lie representation ν^{⋄} : F(L) → End(U). Next, let W^{⋄} be the submodule over En(F(L), ν^{⋄}) generated by h_{1}, …, h_{t}, put

F^{⋄} = (F(L); W^{⋄}) = (F(L); W^{⋄}; ν^{⋄}; h_{1}, …, h_{t}) ,

it is a g.Lie module in 𝔐^{⋄}.

We identify elements in Pol(L⃗, V) with constant functions in U; from proposition 3.6 we have

Ũ ⊆ W^{⋄} .

Observe that all ρ_{i}^{⋄}(a), δ_{i}^{⋄}(a), where i ≤ q, act trivially on constant functions, hence, every submodule of Ũ over En(F(L), ν) (for example, Γ/T (A)) is that of W^{⋄} over En(F(L), ν^{⋄}).

Next, a homomorphism from F to A means a substitution some elements from L in place of generic elements. Hence, it induces a homomorphism from F^{⋄} to A^{⋄} and every homomorphism from F^{⋄} → A^{⋄} is defined by such a substitution. Moreover, for all nonzero g ∈ W^{⋄} there exists a homomorphism to A^{⋄} taking g to a nonzero function. Actually, since g = g(y_{i_{1}}, …, y_{i_{n}}; x_{1}, …, x_{n}) ∈ W^{⋄} is a nonzero function, g(u_{1}, …, u_{n}; x_{1}, …, x_{n}) ≠ 0 for some u_{1}, …, u_{n} ∈ M. It means that under some substitution x → x̄ we have

g(ū_{1}, …, ū_{n}; x̄_{1}, …, x̄_{n}) ≠ 0 .

It unduces a homomorphism from F^{⋄} → A^{⋄} which takes g to g(y_{i_{1}}, …, y_{i_{n}}; x̄_{1}, …, x̄_{n}) ∈ V^{⋄} which does not vanish under the substitution of ū_{1}, …, ū_{n}; therefore, we can identify F^{⋄} with the g.Lie module of generic elements of A^{⋄}.

Now, let

M^{⋄}⟨X⟩ = (L⟨X⟩; V^{⋄}⟨X⟩; v_{1}^{⋄}, …, v_{t}^{⋄})

be the free g.Lie module in 𝔐^{⋄} and let (π, χ) be the epimorphism from M^{⋄} onto F^{⋄} induced by the canonical homomorphism π from L⟨X⟩ onto F(L); then the kernel Ker(χ) is equal to T (A^{⋄}). Put Ξ = χ^{−1} (Γ/T (A)); observe that it is a T-ideal. Indeed, Γ/T (A) is a submodule of W^{⋄} which is closed under the substitution of elements in F(L) in place of generic elements. Since every endomorphism of M^{⋄}⟨X⟩ is defined by such substitution, Ξ is a fully characteristic submodule of V^{⋄}⟨X⟩.

Next, let f be a multilinear element in Ũ, then there exists a multilinear f′ ∈ V^{⋄}⟨X⟩ such that χ(f′) = f; clearly, deg(f′) = deg(f). Now, let f_{n}, n ∈ ℕ, be the appropriate sequence for Γ; since Γ^{#}/T (A) ⊆ Ũ, there is a multilinear f′_{n} ∈ χ^{−1}(f_{n}/T (A)) with deg(f′_{n}) = deg(f_{n}). Clearly, f′_{n} ∉ Ξ and (f′_{n})_{⊤}^{+} ⊆ Ξ for all n ∈ ℕ; so, Ξ is a critical T-ideal. Besides, Ξ^{#} ⊆ Q(A^{⋄}), therefore, T (A^{⋄}) ⊂ Ξ is a “counter-example”.

Proposition 3.8. There exists a finite family of finite dimensional g.Lie modules B^{ξ} ∈ 𝔐^{⋄}, ξ ∈ J, such that

(1)

Ξ ∩ ⋂_{ξ∈J} T (B^{ξ}) ⊆ T (A^{⋄}) ;  (3.78)

(2) type(Ξ, B^{ξ}) < type(Γ, A) for all ξ ∈ J.

Proof. For each α ∈ π_{i}^{∘} there exists a polynomial G^{α} in adjoint operators such that under some substitution of variables x → x̄ it becomes the identity mapping of End(D_{i,α}). Moreover, for all β ∈ ω_{i,α} the L-module W_{i,β} is standard, hence, as a module over D_{i,α}, it is isomorphic to the regular D_{i,α}-module, hence Ḡ^{α} is the identity mapping of Gn(L_{i}^{α}). Moreover, we may assume that G^{α} is linear with respect to at least one variable, y_{α}, and it is the sum of multilinear polynomials. Thus, the set of all homogeneous components of G^{α} consists of multilinear polynomials being linear with respect to y_{α}; we denote this set by H_{α}. Also, put

Y = {y_{α}|α ∈ π^{∘}} ,  (3.79)

where π^{∘} = ⋃_{i=1}^{k} π_{i}^{∘}. Now for each α ∈ π^{∘} take H_{α} ∈ H_{α} and let η denote finite sequence (H^{α}| α ∈ π^{∘}). Take h_{l} ∈ W^{⋄}, see (3.77); since (3.2) holds for all x̄ ∈ V(Γ, A), for every α ∈ π_{i}^{∘} there exists j ∈ {1, …, n} such that μ_{j} is equal to μ^{α} = μ_{i}^{α}, the homomorphism from L onto

L_{i}^{α}/J_{i}^{α} ≃ D_{i,α} ⊕ ⨁_{β∈ω_{i,α}} W_{i,β} .

We fix one of these, so

h_{l} = f_{l}(…, μ_{α}(x), …, μ_{γ}(z), …) ,  (3.80)

where {α, …, γ} is π^{∘}. Next, put

h_{l}^{η} = f_{l}(…, μ_{α}(x · H^{α}), …, μ_{γ}(z · H^{γ}), …) ;

it can be rewritten in the form

h_{l}^{η} = g_{l} · H_{l}^{η} ;

where H_{l}^{η} is a multilinear polynomial in En(M, ν^{⋄}) ⊆ End(U); Observe that varb(H_{l}^{η}) is independent from l and equal to

Y^{η} = ⋃_{α∈π^{∘}} varb(H^{α}) ,

Let α ∈ π_{i}^{∘}, fix a basis of the vector space W_{i,β} for all β ∈ ω_{i,α} and let B_{α} denote the union of these. Take x ∈ Y^{η}, then x ∈ varb(G^{α}) for some α; let x → x̄ be that substitution under which Ḡ^{α} is the identity mapping of D_{i,α}. Besides, Y ⊆ Y^{η}; consider a substitution x → x̃ for elements x ∈ Y^{η} such that

(∘) ỹ_{α} ∈ B_{α} for some variables y_{α} ∈ Y ⊆ Y^{η}, and x̃ = x̄ for the other variables x ∈ Y^{η}. Now we are prepared to define the set of indices J, it consists of pairs ξ = (η, x → x̃),

where η is as above and x → x̃ means a substitution for variables in Y^{η} satisfying (∘); obviously, J is finite. Take ξ = (η, x → x̃) ∈ J and let us define a g.Lie module B^{ξ}. First, let H̃_{l}^{η} denote the value of H_{l}^{η} under the substition x → x̃; it is an operator in En(L, ν^{⋄}) ⊆ End(U). Put

h_{l}^{ξ} = h_{l} · H̃_{l}^{η} ,

and let V^{ξ} be the submodule of U generated by h_{1}^{ξ}, …, h_{t}^{ξ} over En(L, ν^{⋄}); then

B^{ξ} = (L; V^{ξ}; ν^{⋄}; h_{1}^{ξ}, …, h_{t}^{ξ})

is a finite dimensional g.Lie module in 𝔐^{⋄}.

Lemma 3.26. For all ξ ∈ J

type(Ξ, B^{ξ}) < type(Γ, A) .

Proof. By irreducibility of T (A) ⊂ Γ, ∥a(Ξ, B^{ξ})∥ ≤ ∥a(Γ, A)∥, a(Ξ, B^{ξ}) ≤ a(Γ, A), and we may assume that these are equalities (otherwise, the lemma is proven).

Next, let us show that t(Ξ, B^{ξ}) ≤ t(Γ, A); let a = a(Γ, A) = a(Ξ, B^{ξ}), take some b ∈ (ℕ_{0}^{∞})^{∞}, c_{i}, d_{i} ∈ ℕ and put

S = S_{a,b,c} .

Suppose that S(Ξ) ⊈ T (B^{ξ}), take a typical generator h = S(w) ∉ T (B^{ξ}), then g = χ(h) ≠ 0. By defintion, w = ∑_{l=1}^{t} v_{l}^{⋄} · E_{l} for some E_{l} ∈ En(M^{⋄}⟨X⟩) and χ(w) ∈ Υ̃, hence

χ(w) = ∑_{l=1}^{t} ∑_{u_{1},... ,u_{n},E} f_{l}(μ_{1}(u_{1}), …, μ_{n}(u_{n})) · E ,

where u_{1}, …, u_{n} ∈ F(L) and E ∈ En(M, ν).

Next, since χ is induced by the identity mapping, we have χ(h) = χ(S(w)) = S(χ(w)). Take l, u_{1}, …, u_{n}, E such that

g = S(f_{l}(μ_{1}(u_{1}), …, μ_{n}(u_{n})) · E)  (3.81)

differs from 0; for all j there exist i, γ such that μ_{j} = μ_{i}^{γ}; as μ_{j} kills all elements from L_{i′}, i′ ≠ i, we can assume that u_{j} ∈ F(L)_{i}.

Recall that for all α ∈ π_{i}^{∘} ⊆ π^{∘} we have an operator G^{α} = G_{i}^{α} such that under some substitution of elements in D_{i,α} the operator Ḡ_{i}^{α} is equal to the identity mapping of L_{i}^{α}/J_{i}^{α}; also, it acts trivially on the others direct summands of decomposition (3.46), so it is equal to the projector from L_{i}/J_{i} onto L_{i}^{α}/J_{i}^{α}. Also, put G_{i}^{•} = 1 − ∑_{α∈π_{i}^{∘}} G_{i}^{α}, where 1 is the identity mapping of L_{i}/J_{i}, then Ḡ_{i}^{•} is the projector onto L_{i}^{•}/J_{i}^{•}. Hence, the function (3.81) can be written as

g = S(f_{l}(μ_{1}(u_{1}) · Ḡ_{1}, …, μ_{n}(u_{n}) · Ḡ_{n}) · E)
= S(f_{l}(u_{1} · Ḡ_{1}, …, u_{n} · Ḡ_{n}) · E) ,

where G_{j} = G_{i}^{γ} provided μ_{j} = μ_{i}^{γ}. We can choose G_{j} being independent from variables of g, hence, the element above is obtained from

S(f_{l}(u_{1} · G_{1}, …, u_{n} · G_{n}) · E) ∈ S(Γ^{#})/T (A)

by substitution x → x̄ for all x ∈ varb(G_{j}); that means that S(Γ^{#}) ⊈ T (A) as required.

Let us show that |b(Ξ, B^{ξ})| < |b(Γ, A)|; take x̄ ∈ V(f, B^{ξ}) , where f ∈ Ξ. Then f(x̄) is a nonzero function

g = ∑_{i_{1},... ,i_{m}} g_{i_{1},... ,i_{m}}(y_{i_{1}}, …, y_{i_{m}}) ;

there exist x̄_{i_{j}} ∈ ⋃_{i=1}^{k} L_{i} such that

ḡ = ∑_{i_{1},... ,i_{m}} g_{i_{1},... ,i_{m}}(x̄_{i_{1}}, …, x̄_{i_{m}}) ≠ 0 .

We can assume that all x_{i_{j}} ∈ X are not in varb(f), so the substitution x_{i_{j}} → x̄_{i_{j}} extends x̄. Hence, ḡ = f̄′, where

f′ = f · δ_{i_{1}}(x_{i_{1}}) … δ_{i_{m}}(x_{i_{m}}) ∈ Ξ ,

thus, we can replace f with f′ and assume that f̄ is a constant, f̄ ∈ V ⊆ V^{⋄}.

On the other hand, f = ∑_{l=1}^{t} v_{l}^{⋄} · E_{l}, hence

f(x̄) = ∑_{l=1}^{t} h_{l}^{ξ} · Ē_{l} = ∑_{l=1}^{t} h_{l} · H̃_{l}^{η}Ē_{l} .

Take l such that h_{l} · H̃_{l}^{η}Ē_{l} ≠ 0; by definition, g = h_{l} · H_{l}^{η}E_{l} is equal to

∑_{u_{1},... ,u_{n},E} f_{l}(μ_{1}(u_{1}), …, μ_{n}(u_{n})) · E ,

where u_{1}, …, u_{n} ∈ M = F(L) and E ∈ En(M, ν). We may assume that H_{l}^{η} and E_{l} depend on disjoint set of variables, hence g is multilinear and

varb(g) = varb(E) ∪ ⋃_{i=1}^{n} varb(u_{i}) = varb(H_{l}^{η}) ∪ varb(E_{l}) .  (3.82)

Observe that the joint substitution, x → x̂, of x → x̃ (for variables of H_{l}^{ξ}) and x → x̄ (for E_{l}) takes some f_{l}(μ_{1}(u_{1}), …, μ_{n}(u_{n})) to a nonzero element,

f_{l}(μ_{1}(û_{1}), … μ_{n}(û_{n})) .

Since f_{l} ∈ Γ^{#}, the total number of variables in (3.82) such that x̂ ∈ W is equal to |b(Γ, A)|. Also, by definition, there exists x ∈ varb(H_{l}^{η}) such that x̃ ∈ W, hence |b(x̄)| < |b(Γ, A)|. □

So, part (2) of the proposition is proven. Let I denote the left hand side of (3.78) and suppose that it does not hold. Take x̄ ∈ V(w, A^{⋄}) for some w ∈ I ⊆ Ξ. Hence, 0 ≠ w(x̄) = χ(w)(x̄), where χ(w) ∈ Γ, hence x̄ ∈ V(Γ, A). On the other hand, w = ∑_{j=1}^{t} v_{l}^{⋄} · E_{l} and

χ(w) = ∑_{l=1}^{t} h_{l} · E_{l} .

Consider the operator polynomials G^{α}, α ∈ π^{∘}, defined above; recall that Ḡ_{α} is the identity mapping of L_{i}^{α}/J_{i}^{α}. Using (3.80), we can define an operator G_{l} ∈ En(L⟨X⟩, ν^{⋄}) so that

h_{l} · G_{l} = h_{l}(…, μ_{α}(x · G^{α}), …, μ_{γ}(z · G^{γ}), …) .

Under the substitution x → x̄ for variables of G_{α} for all α, we get Ḡ_{l} which is equal to the identity mapping, hence,

χ(w)(x̄) = ∑_{l=1}^{t} g_{l} · Ḡ_{l}Ē_{l} .

Observe that G_{l} is the sum of homogeneous components H_{l}^{η}, hence, there is η such that the same substitution gives

∑_{l=1}^{t} h_{l} · H̄_{l}^{η}Ē_{l} ≠ 0 .

Put h = ∑_{l=1}^{t} h_{l} · H_{l}E_{l} ; obviously, we can assume that H_{l} and E_{l} depend on different variables, hence h is multilinear. Moreover, ξ(w) ∈ Γ/T (A) ⊆ Ũ is a constant function, hence

h = ∑_{l=1}^{t} ∑_{(u_{1},... ,u_{n}),E} f_{l}(μ_{1}(u_{1}), …, μ_{n}(u_{n})) · E

where (u_{1}, …, u_{n}) is an n-tuple of multilinear Lie polynomials and E ∈ En(F(L), ν). Although the parameters of the second sum, u_{1}, …, u_{n}, E, depends on l as well as the number n and μ_{1}, …, μ_{n}, the set of variables varb(E)∪⋃_{j=1}^{n} varb(u_{j}) is independent from l and equal to varb(H_{l}) ∪ varb(E_{l}) = varb(h) for all l.

Recall that each μ_{j} is the homomorphism μ_{i}^{γ} from L onto L_{i}^{γ}/J_{i}^{γ} for some γ ∈ π̃_{i}^{∘}. If there is x ∈ varb(u_{j}) such that x̄ ∉ L_{i}^{γ}, then μ_{j}(ū_{j}) = 0. Otherwise μ_{j}(ū_{j}) = ū_{j}, hence

h̄ = ∑_{l=1}^{t} ∑_{u′_{1},... ,u′_{n},E} h_{l}(ū′_{1}, …, ū′_{n}) · Ē ,

where (u′_{1}, …, u′_{n}) runs other the subset of all (u_{1}, …, u_{n}) satisfying the following condition for all l :

(*) x̄ ∈ L_{i}^{γ} for all x ∈ varb(u_{j}) provided μ_{j} = μ_{i}^{γ}.

Now we are prepared to use Main Lemma; put

f = ∑_{l=1}^{t} ∑_{(u′_{1},... ,u′_{n},E)} f_{l}(u′_{1}, …, u′_{n}) · E ,

it is a multilinear polynomial which does not vanish under the substitution x → x̄. Subset of variables (3.79) plays the role of Y from Main Lemma:

Y ⊆ varb(H_{l}^{η}) ⊆ ⋃_{j=1}^{n} varb(u′_{j})

for all l and (u′_{1}, …, u′_{n}), hence, Y ⊆ varb(f). Also, the subset Z = Z(f, x → x̄), see (3.53), is contained in varb(E_{l}) for all l, because each x ∈ varb(H_{l}^{η}) belongs to varb(G^{α}) for some α ∈ π_{i}^{∘} and, therefore, takes value in D_{i,α}.

By Main lemma there exists a substitution x → x̃ such that

(a) if x̄ ∈ L_{i}^{γ}, then x̃ ∈ L_{i}^{γ} for all x ∈ varb(f);

(b) if x ∉ Y ∪ Z, then x̃ = x̄;

(c) there exist y = y_{α} ∈ Y such that ỹ belongs to W_{i,β} for some β ∈ ω_{i,α} .

From (a) it follows that for all j we have μ_{j}(ũ_{j}) = ũ_{j}, if u_{j} satisfies (*), and μ_{j}(ũ_{j}) = 0, otherwise. Under this substitution

h̃ = ∑_{l=1}^{t} ∑_{u_{1},... ,u_{n}} f_{l}(μ_{1}(ũ_{1}), …, μ_{n}(ũ_{n})) · Ẽ =

= ∑_{l=1}^{t} ∑_{u′_{1},... ,u′_{n})} f_{l}(ũ′_{1}, …, ũ′_{n}) · Ẽ ≠ 0 .

In other terms, h̃ = ∑_{l=1}^{t} h_{l} · H̃_{l}^{η}Ẽ_{l} ≠ 0; since h is linear with respect to each y_{α} ∈ Y, we may assume that ỹ_{α} ∈ B_{α}, see (∘). It means that w ∉ T (B^{ξ}), where ξ = (η, x → x̃); indeed, its value under the substitution x → x̃ for all x ∈ varb(w) = varb(E_{l}) is equal to

∑_{l=1}^{t} h_{l}^{ξ} · Ẽ_{l} = h̃ ≠ 0 ,

which contradicts to the choice of w. Therefore, the proposition is proven. □

Thus,

Ξ ∩ ⋂_{ξ∈J} T (B^{ξ}) = T (A′) ∩ ⋂_{ξ∈J} T (B^{ξ}) ;

the right hand side is the ideal of identities of the finite dimensional g.Lie module

A′ ⊕ ⨁_{ξ∈J} B^{ξ} ;

besides, Ξ ⊆ Q(B^{ξ}) for all ξ. Hence, by proposition 2.15 there must exist a ”counterexample” of lesser type, which is impossible. Therefore, ”counter-examples” do not exist; theorem 2.1 is proven.

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