Yangians and transvector algebras

A. I. Molev

School of Mathematics and Statistics University of Sydney, NSW 2006, Australia alexm@maths.usyd.edu.au

November 1998

Abstract

Olshanski’ s centralizer construction provides a realization of the Yangian Y(m) for the Lie algebra 𝔤𝔩(m) as a subalgebra in the projective limit algebra A_{m} = lim proj A_{m}(n) as n → ∞, where A_{m}(n) is the centralizer of 𝔤𝔩(n−m) in the enveloping algebra U(𝔤𝔩(n)). We give a modified version of this construction based on a quantum analog of Sylvester’s theorem. We then use it to get an algebra homomorphism from the Yangian Y(n) to the transvector algebra associated with the pair 𝔤𝔩(m) ⊂ 𝔤𝔩(m + n). The results are applied to identify the elementary representations of the Yangian by constructing their highest vectors explicitly in terms of elements of the transvector algebra.

0 Introduction

Let A_{m}(n) denote the centralizer of 𝔤𝔩(n − m) in the universal enveloping algebra U(𝔤𝔩(n)). In particular, A_{0}(n) is the center of U(𝔤𝔩(n)). It was shown by Olshanski [20, 21] that for any fixed m there exists a chain of natural homomorphisms

A_{m}(m) ← A_{m}(m + 1) ← ⋯ ← A_{m}(n) ← ⋯  (0.1)

and one can define the corresponding projective limit algebra

A_{m} = lim proj A_{m}(n),  n → ∞.  (0.2)

The algebra A_{0} is isomorphic to an algebra of polynomials in countably many variables while for m > 0 one has the tensor product decomposition [20, 21]:

A_{m} = A_{0} ⊗ Y(m),  (0.3)

where Y(m) = Y(𝔤𝔩(m)) is the Yangian for the Lie algebra 𝔤𝔩(m); see the definition in Section 1 below. The algebra Y(m) first appeared in the works of Faddeev’s school on the Yang–Baxter equation; see e.g. [11, 12, 23]. The Yangian Y(m) is a Hopf algebra which can be regarded as a deformation of the enveloping algebra U(𝔤𝔩(m)[x]), where 𝔤𝔩(m)[x] is the Lie algebra of 𝔤𝔩(m)-valued polynomials [5].

The key part of the proof of the decomposition (0.3) is a construction of algebra homomorphisms

Y(m) → A_{m}(n),  n = m, m + 1, …  (0.4)

compatible with the chain (0.1). Then one shows that this defines an embedding

Y(m) ↪ A_{m}  (0.5)

so that the Yangian Y(m) can be identified with a subalgebra in A_{m}; see [20, 21].

Note that a projective limit algebra of type (0.2) can be also constructed for the series of the orthogonal and symplectic Lie algebras. This limit algebra turns out to be related with the corresponding twisted Yangians; see [15, 22] for more details.

In this paper we construct a new family of homomorphisms (0.4) which define a new embedding (0.5). Then we give a modified tensor product decomposition of type (0.3). Our argument is based on a quantum analog of Sylvester’s theorem for the Yangians (Theorem 1.3); cf. [8, 10].

We also use Theorem 1.3 to identify the so-called elementary representations of Y(n). They naturally arise from the centralizer construction as follows. Consider a finite-dimensional irreducible representation L(λ) of the Lie algebra 𝔤𝔩(m + n) with the highest weight λ and denote by L(λ)_{μ}^{+} the subspace in L(λ) of 𝔤𝔩(m)-highest vectors of weight μ. It is well-known (see e.g. [4, Section 9.1]) that L(λ)_{μ}^{+} is an irreducible representation of the centralizer A = U(𝔤𝔩(m + n))^{𝔤𝔩(m)}. We can make L(λ)_{μ}^{+} into a Y(n)-module via the homomorphism Y(n) → A; cf. (0.4). This module can be shown to be irreducible; see Section 4.1.

On the other hand, the subspace L(λ)^{+} of 𝔤𝔩(m)-highest vectors in L(λ) is preserved by the so-called raising and lowering operators. The latter generate the transvector algebra Z associated with the pair 𝔤𝔩(m) ⊂ 𝔤𝔩(m + n) (which is sometimes called the S-algebra or Mickelsson algebra); see [26]–[28]. By definition of Z (see Section 3) there is a natural algebra homomorphism A → Z so that the action of generators of Y(n) in L(λ)_{μ}^{+} can be explicitly expressed in terms of the lowering and raising operators (see Theorem 3.1).

We explicitly construct the highest vector of the Y(n)-module L(λ)_{μ}^{+} in terms of the lowering operators and calculate its highest weight. We also identify L(λ)_{μ}^{+} as a Y(𝔰𝔩(n))-module by calculating its Drinfeld polynomials. They turn out to be the same as those found by Nazarov and Tarasov [17] for a different representation of Y(𝔰𝔩(n)) in L(λ)_{μ}^{+} defined via the Olshanski homomorphism (0.4), which shows that these two Y(𝔰𝔩(n))-modules are isomorphic.

Following [17] we call the Y(n)-module L(λ)_{μ}^{+} elementary. These representations play an important role in the classification of the Y(n)-modules with a semisimple action of the Gelfand–Tsetlin subalgebra; see [3, 17]. In particular, it was proved in [17, Theorem 4.1] that, up to an automorphism of Y(n), any such module is isomorphic to a tensor product of elementary representations.

1 Quantum Sylvester’s theorem

A detailed description of the algebraic structure of the Yangian for the Lie algebra 𝔤𝔩(n) is given in the expository paper [14]. In this section we reproduce some of those results and use them to prove a quantum analog of Sylvester’s theorem.

The Yangian Y(n) = Y(𝔤𝔩(n)) is the complex associative algebra with the generators t_{ij}^{(1)}, t_{ij}^{(2)}, … where 1 ≤ i, j ≤ n, and the defining relations

[t_{ij}(u), t_{kl}(v)] = \frac{1}{u − v} (t_{kj}(u)t_{il}(v) − t_{kj}(v)t_{il}(u)),  (1.1)

where

t_{ij}(u) = δ_{ij} + t_{ij}^{(1)}u^{−1} + t_{ij}^{(2)}u^{−2} + ⋯ ∈ Y(n)[[u^{−1}]]

and u is a formal variable. Introduce the matrix

T (u) := ∑_{i,j=1}^{n} t_{ij}(u) ⊗ E_{ij} ∈ Y(n)[[u^{−1}]] ⊗ End ℂ^{n},

where the E_{ij} are the standard matrix units. Then the relations (1.1) are equivalent to the single relation

R(u − v)T_{1}(u)T_{2}(v) = T_{2}(v)T_{1}(u)R(u − v).  (1.2)

Here T_{1}(u) and T_{2}(u) are regarded as elements of Y(n)[[u^{−1}]] ⊗ End ℂ^{n} ⊗ End ℂ^{n}, the subindex of T (u) indicates to which copy of End ℂ^{n} this matrix corresponds, and

R(u) = 1 − P u^{−1}, P = ∑_{i,j=1}^{n} E_{ij} ⊗ E_{ji} ∈ (End ℂ^{n})^{⊗2}.

The quantum determinant qdet T (u) of the matrix T (u) is a formal series in u^{−1} with coefficients from Y(n) defined by

qdet T (u) = ∑_{p∈𝔖_{n}} sgn(p) t_{p(1)1}(u) ⋯ t_{p(n)n}(u − n + 1).  (1.3)

The coefficients of the quantum determinant qdet T (u) are algebraically independent generators of the center of the algebra Y(n).

We shall need a generalization of the relation (1.2) for elements of multiple tensor products of the form Y(n)[[u^{−1}]]⊗End ℂ^{n}⊗⋯⊗End ℂ^{n}. For an operator X ∈ End ℂ^{n} and a number s = 1, 2, … we set

X_{i} := 1^{⊗(i−1)} ⊗ X ⊗ 1^{⊗(s−i)} ∈ (End ℂ^{n})^{⊗s}, 1 ≤ i ≤ s.

Similarly, if X ∈ (End ℂ^{n})^{⊗2} then for any i, j such that 1 ≤ i, j ≤ s and i ≠ j, we denote by X_{ij} the operator in (ℂ^{n})^{⊗s} which acts as X in the product of ith and jth copies and as 1 in all other copies. Let u_{1}, …, u_{s} be formal variables. Set

R(u_{1}, …, u_{s}) := (R_{s−1,s})(R_{s−2,s}R_{s−2,s−1}) ⋯ (R_{1s} ⋯ R_{12}),  (1.4)

where we abbreviate R_{ij} := R_{ij}(u_{i} − u_{j}). Note that the following (Yang–Baxter) relation is satisfied by the R_{ij}

R_{ij}R_{ir}R_{jr} = R_{jr}R_{ir}R_{ij},  (1.5)

and the factors R_{ij} and R_{rs} with distinct indices are permutable in (1.4). This allows one to deduce the following identity from (1.2):

R(u_{1}, …, u_{s}) T_{1}(u_{1}) ⋯ T_{s}(u_{s}) = T_{s}(u_{s}) ⋯ T_{1}(u_{1}) R(u_{1}, …, u_{s}).  (1.6)

The symmetric group 𝔖_{s} naturally acts in the tensor space (ℂ^{n})^{⊗s} by permutations of the tensor factors. If the variables in (1.4) satisfy the condition u_{i} − u_{i+1} = 1 for i = 1, …, s − 1 then R(u_{1}, …, u_{s}) becomes the antisymmetrization operator in (ℂ^{n})^{⊗s}:

R(u_{1}, …, u_{s}) = A_{s} := ∑_{q∈𝔖_{s}} sgn(q) Q,  (1.7)

where Q is the operator in (ℂ^{n})^{⊗s} corresponding to a permutation q ∈ 𝔖_{s}. By (1.6) we then have

A_{s} T_{1}(u) ⋯ T_{s}(u − s + 1) = T_{s}(u − s + 1) ⋯ T_{1}(u) A_{s}.  (1.8)

The operator (1.8) can be written in the form

∑ t _{b_{1}··· b_{s}}^{a_{1}··· a_{s}}(u) ⊗ E_{a_{1}b_{1}} ⊗ ⋯ ⊗ E_{a_{s}b_{s}},

summed over the indices a_{i}, b_{i} ∈ {1, …, n}, where t _{b_{1}··· b_{s}}^{a_{1}··· a_{s}}(u) ∈ Y(n)[[u^{−1}]]. In particular, t _{b}^{a}(u) = t_{ab}(u). Note that by (1.8) the series t _{b_{1}··· b_{s}}^{a_{1}··· a_{s}}(u) is antisymmetric with respect to permutations of the upper indices and of the lower indices. It can be given by the following explicit formulas

t _{b_{1}··· b_{s}}^{a_{1}··· a_{s}}(u) = ∑_{σ∈𝔖_{s}} sgn(σ) t_{a_{σ(1)}b_{1}}(u) ⋯ t_{a_{σ(s)}b_{s}}(u − s + 1)  (1.9)
= ∑_{σ∈𝔖_{s}} sgn(σ) t_{a_{1}b_{σ(1)}}(u − s + 1) ⋯ t_{a_{s}b_{σ(s)}}(u).  (1.10)

Note that t _{1··· n}^{1··· n}(u) = qdet T (u); see (1.3).

Proposition 1.1 We have the relations

[t _{b_{1}··· b_{k}}^{a_{1}··· a_{k}}(u), t _{d_{1}··· d_{l}}^{c_{1}··· c_{l}}(v)] = ∑_{p=1}^{min\{k,l\}} \frac{(−1)^{p−1} p!}{(u − v − k + 1) ⋯ (u − v − k + p)}

∑_{{i_{1}<···<i_{p} \atop j_{1}<···<j_{p}}} (t _{b_{1} ··· b_{k}}^{a_{1}··· c_{j_{1}}··· c_{jp}··· a_{k}}(u)t _{d_{1} ··· d_{l}}^{c_{1}··· a_{i_{1}}··· a_{ip}··· c_{l}}(v) − t _{d_{1}··· b_{i_{1}}··· b_{ip}··· d_{l}}^{c_{1} ··· c_{l}}(v)t _{b_{1}··· d_{j_{1}}··· d_{jp}··· b_{k}}^{a_{1} ··· a_{k}}(u)) .

Here the p-tuples of upper indices (a_{i_{1}}, …, a_{i_{p}}) and (c_{j_{1}}, …, c_{j_{p}}) are respectively interchanged in the first summand on the right hand side while the p-tuples of lower indices (b_{i_{1}}, …, b_{i_{p}}) and (d_{j_{1}}, …, d_{j_{p}}) are interchanged in the second summand.

Proof. Let us take s = k + l in (1.6) and specialize the variables u_{i} as follows:

u_{i} = u − i + 1, i = 1, …, k and u_{k+j} = v − j + 1, j = 1, …, l.  (1.11)

Using (1.5) we can bring the product (1.4) to the form

R(u_{1}, …, u_{k+l}) = ( ∏_{j=1,..., l}^{→} R_{k,k+j} ⋯ R_{1,k+j}) R(u_{1}, …, u_{k}) R(u_{k+1}, …, u_{k+l}). (1.12)

However, by (1.7) and (1.11)

R(u_{1}, …, u_{k}) = A_{k},  R(u_{k+1}, …, u_{k+l}) = A′_{l},  (1.13)

where A′_{l} is the antisymmetrizer corresponding to the set of indices {k + 1, …, k + l}. Let us show that the expression (1.12) can then be written as

R̃(u, v) A_{k}A′_{l}  (1.14)

with

R̃(u, v) = ∑_{p=0}^{min\{k,l\}} \frac{(−1)^{p} p!}{(u − v − k + 1) ⋯ (u − v − k + p)} ∑_{{i_{1}<···<i_{p} \atop j_{1}<···<j_{p}}} P_{i_{1},k+j_{1}} ⋯ P_{i_{p},k+j_{p}}.

(1.15)

Indeed, if indices i_{1}, …, i_{s} ∈ {1, …, k} are distinct then

P_{i_{1},k+j} ⋯ P_{i_{s},k+j} A_{k} = (−1)^{s−1}P_{i_{s},k+j} A_{k}.

This implies that for each j

R_{k,k+j} ⋯ R_{1,k+j} A_{k} = (1 − \frac{P_{1,k+j} + ⋯ + P_{k,k+j}}{u − v − k + j} ) A_{k}.  (1.16)

Similarly, if indices j_{1}, …, j_{s} ∈ {1, …, l} are distinct then

P_{i,k+j_{1}} ⋯ P_{i,k+j_{s}} A′_{l} = (−1)^{s−1}P_{i,k+j_{s}} A′_{l}.  (1.17)

Moreover, if i_{1} ≠ i_{2} and j_{1} ≠ j_{2} then

P_{i_{1},k+j_{1}}P_{i_{2},k+j_{2}} A_{k}A′_{l} = P_{i_{1},k+j_{2}}P_{i_{2},k+j_{1}} A_{k}A′_{l}.

Using this relation together with (1.17) we bring (1.12) to the form (1.14) by a straightforward calculation. Now (1.6) takes the form

R̃(u, v)A_{k}T_{1}(u) ⋯ T_{k}(u − k + 1) A′_{l} T_{k+1}(v) ⋯ T_{k+l}(v − l + 1)
= A′_{l} T_{k+1}(v) ⋯ T_{k+l}(v − l + 1) A_{k}T_{1}(u) ⋯ T_{k}(u − k + 1)R̃(u, v), (1.18)

where we have used the identity (1.8) and the fact that R̃(u, v) A_{k}A′_{l} = A_{k}A′_{l} R̃(u, v) which is easy to verify. To complete the proof we take the coefficients at E_{a_{1}b_{1}} ⊗ ⋯ ⊗ E_{a_{k}b_{k}} ⊗ E_{c_{1}d_{1}} ⊗ ⋯ ⊗ E_{c_{l}d_{l}} on both sides of (1.18) (or, equivalently, apply both sides to the basis vector e_{b_{1}} ⊗ ⋯ ⊗ e_{b_{k}} ⊗ e_{d_{1}} ⊗ ⋯ ⊗ e_{d_{l}} of (ℂ^{n})^{⊗(k+l)} and compare the coefficients at the vector e_{a_{1}} ⊗ ⋯ ⊗ e_{a_{k}} ⊗ e_{c_{1}} ⊗ ⋯ ⊗ e_{c_{l}}, where e_{1}, …, e_{n} is the canonical basis of ℂ^{n}).

Introduce the quantum comatrix T̂(u) = (t̂_{ij}(u)) for the matrix T (u) by

t̂_{ij}(u) = (−1)^{i+j}t _{1 ··· î··· n}^{1 ··· ĵ··· n}(u),

where the hats on the right hand side indicate the indices to be omitted. Equivalently, T̂(u) can be defined by

A_{n}T_{1}(u) ⋯ T_{n−1}(u − n + 2) = A_{n}T̂_{n}(u).

Taking s = n in (1.8) we derive the identity

T̂(u)T (u − n + 1) = qdet T (u).  (1.19)

The Poincaré–Birkhoff–Witt theorem for the algebra Y(n) (see e.g. [14, Corollary 1.23]) implies that for m ≤ n the Yangians Y(m) and Y(n − m) can be identified with the subalgebras in Y(n) generated by the coefficients of the series t_{ij}(u) with i, j ≤ m and m < i, j ≤ n, respectively. For any indices i, j ≤ m introduce the following series with coefficients in Y(n)

t̃_{ij}(u) = t _{j,m+1··· n}^{i,m+1··· n}(u)  (1.20)

and combine them into the matrix T̃(u) = (t̃_{ij}(u)). For subsets P and Q in {1, …, n} and an n × n-matrix X we shall denote by X_{PQ} the submatrix of X whose rows and columns are enumerated by P and Q respectively. Set A = {1, …, m} and B = {m + 1, …, n}. We shall need the following generalization of (1.19).

Proposition 1.2 We have the identity

T̂(u)_{AA} T̃(u − m + 1) = qdet T (u) qdet T (u − m + 1)_{BB}.  (1.21)

Proof. For m = 1 the relation (1.21) is trivial. We shall assume that m ≥ 2. Consider the identity (1.6) with s = 2n − m and specialize the variables u_{i} as follows

u_{i} = u − i + 1, i = 1, …, n − 1 and u_{n+j−1} = v − j + 1, j = 1, …, n − m + 1.

We shall show that (1.6) then becomes a rational function in v with a simple pole at v = u − m + 1 and calculate the corresponding residue in two different ways. First, write (1.6) in the form (1.18) with k = n − 1 and l = n − m + 1. Then fix indices i, j ∈ {1, …, m} and apply the left hand side of (1.18) to the basis vector

e_{1} ⊗ ⋯ ⊗ e_{i−1} ⊗ e_{i+1} ⊗ ⋯ ⊗ e_{n} ⊗ e_{j} ⊗ e_{m+1} ⊗ ⋯ ⊗ e_{n}  (1.22)

in (ℂ^{n})^{⊗(2n−m)} and take the coefficient at the vector

e_{1} ⊗ ⋯ ⊗ e_{i−1} ⊗ e_{i+1} ⊗ ⋯ ⊗ e_{n} ⊗ e_{i} ⊗ e_{m+1} ⊗ ⋯ ⊗ e_{n}.  (1.23)

A straightforward calculation shows that this matrix element equals

\frac{u − v − n + 1}{u − v − m + 1} (t̂_{ii}(u)t̃_{ij}(v) + \frac{1}{u − v − m + 2} ∑_{a=1, a≠i}^{m} t̂_{ia}(u)t̃_{aj}(v)) .

Multiplying this expression by u − v − m + 1 and setting v = u − m + 1 we obtain

(m − n) (T̂(u)_{AA} T̃(u − m + 1))_{ij} .

On the other hand, transform R(u_{1}, …, u_{2n−m}) applying (1.12) and (1.13) with k = n − 1 and l = n − m + 1. Using (1.16) we can write R(u_{1}, …, u_{2n−m}) as

( ∏_{j=1,..., l−1}^{→} R_{k,k+j} ⋯ R_{1,k+j}) (1 − \frac{P_{1,k+l} + ⋯ + P_{k,k+l}}{u − v − k + l} ) A_{k}A′_{l}.  (1.24)

Note that (l − 1)! A′_{l} = A′_{l−1}A′_{l}. Therefore, repeating the argument of the proof of Proposition 1.1 we can transform (1.24) to get the expression

\frac{1}{(l − 1)!} R̃(u, v)A_{k}A′_{l−1} (1 − \frac{P_{1,k+l} + ⋯ + P_{k,k+l}}{u − v − k + l} ) A′_{l},  (1.25)

where R̃(u, v) is given by (1.15) with l replaced by l − 1. Finally, put k = n − 1, l = n − m + 1 into (1.25), multiply it by u − v − m + 1 and then put v = u − m + 1. This gives

(m − n) ∑ P_{i_{1},n} ⋯ P_{i_{n−m},2n−m−1}(1 − P_{1,2n−m} − ⋯ − P_{n−1,2n−m})A_{n−1}A′_{n−m+1}, (1.26)

summed over the sets of indices 1 ≤ i_{1} < ⋯ < i_{n−m} ≤ n − 1. Note that

Ã_{n} := (1 − P_{1,2n−m} − ⋯ − P_{n−1,2n−m})A_{n−1}

is the antisymmetrizer corresponding to the subset of indices {1, …, n − 1, 2n − m}. The application of the operator (1.26) to the vector (1.22) gives the zero vector unless i = j since it is annihilated by Ã_{n}A′_{n−m+1}. Suppose now that i = j. By (1.8) we can write

A′_{n−m+1}T_{n}(u − m + 1) ⋯ T_{2n−m}(u − n + 1) =
T_{2n−m}(u − n + 1) ⋯ T_{n}(u − m + 1)A′_{n−m+1}.

The residue of the left hand side of (1.6) at v = u − m + 1 then takes the form

(m − n) ∑ P_{i_{1},n} ⋯ P_{i_{n−m},2n−m−1}Ã_{n}T_{1}(u) ⋯ T_{n−1}(u − n + 2)T_{2n−m}(u − n + 1)
T_{2n−m−1}(u − n + 2) ⋯ T_{n}(u − m + 1)A′_{n−m+1}.

The diagonal matrix element of this operator corresponding to the vector (1.23) equals

(m − n) qdet T (u) qdet T (u − m + 1)_{BB}

which completes the proof.

We are now in a position to prove a quantum analog of Sylvester’s theorem for the Yangians (Theorem 1.3). An analog of Sylvester’s theorem for the quantized algebra of functions on GL(n) was given by Krob and Leclerc [10] with the use of the quasi-determinant version of this theorem due to Gelfand and Retakh [8]. As was remarked in [10], the same approach can be used for the case of the Yangians. We give a different proof based on Propositions 1.1 and 1.2. We use notation (1.20).

Theorem 1.3 The mapping

t_{ij}(u) ↦ t̃_{ij}(u),  1 ≤ i, j ≤ m  (1.27)

defines an algebra homomorphism Y(m) → Y(n). Moreover, one has the identity

qdet T̃(u) = qdet T (u) qdet T (u − 1)_{BB} ⋯ qdet T (u − m + 1)_{BB}. (1.28)

Proof. To check that the elements (1.20) satisfy the defining relations (1.1) we use Proposition 1.1. This proves the first part of the theorem.

To prove (1.28) we use induction on m. For m = 1 the identity is trivial. Let m ≥ 2. We obtain from (1.19) that

T̂(u) = qdet T (u) T (u − n + 1)^{−1}.

Taking the mmth entry on both sides we obtain

qdet T (u)_{CC} = qdet T (u) (T (u − n + 1)^{−1})_{mm} ,  (1.29)

where C = {1, …, m − 1, m + 1, …, n}. Similarly, using the homomorphism (1.27) we can write the corresponding analog of (1.19) for the matrix T̃(u) which gives

qdet T̃(u)_{A′A′} = qdet T̃(u) (T̃(u − m + 1)^{−1})_{mm} ,  (1.30)

where A′ = {1, …, m − 1}. By the induction hypothesis,

qdet T̃(u)_{A′A′} = qdet T (u)_{CC} qdet T (u − 1)_{BB} ⋯ qdet T (u − m + 2)_{BB}.

It remains to apply (1.29), (1.30) and the identity

(T (u − n + 1)^{−1})_{AA} = T̃(u − m + 1)^{−1} qdet T (u − m + 1)_{BB},

which is a corollary of (1.19) and (1.21).

2 Modified centralizer construction

We start with the definition of the Olshanski algebra A_{m}; see [20, 21]. Fix a nonnegative integer m and for any n ≥ m denote by 𝔤_{m}(n) the subalgebra in the general linear Lie algebra 𝔤𝔩(n) spanned by the basis elements E_{ij} with m + 1 ≤ i, j ≤ n.

The subalgebra 𝔤_{m}(n) is isomorphic to 𝔤𝔩(n − m). Let A_{m}(n) denote the centralizer of 𝔤_{m}(n) in the universal enveloping algebra A(n) := U(𝔤𝔩(n)). In particular, A_{0}(n) is the center of U(𝔤𝔩(n)). Let A(n)^{0} denote the centralizer of E_{nn} in A(n) and let I(n) be the left ideal in A(n) generated by the elements E_{in}, i = 1, …, n. Then I(n)^{0} := I(n) ∩ A(n)^{0} is a two-sided ideal in A(n)^{0} and one has a vector space decomposition

A(n)^{0} = I(n)^{0} ⊕ A(n − 1).

Therefore the projection of A(n)^{0} onto A(n − 1) with the kernel I(n)^{0} is an algebra homomorphism. Its restriction to the subalgebra A_{m}(n) defines a filtration preserving homomorphism

π_{n} : A_{m}(n) → A_{m}(n − 1)  (2.1)

so that one can define the algebra A_{m} as the projective limit (0.2) in the category of filtered algebras. In other words, an element of the algebra A_{m} is a sequence of the form a = (a_{m}, a_{m+1}, …, a_{n}, …) where a_{n} ∈ A_{m}(n), π_{n}(a_{n}) = a_{n−1} for n > m, and

deg a := sup_{n≥m} deg a_{n} < ∞,

where deg a_{n} denotes the degree of a_{n} in the universal enveloping algebra A(n).

The algebra A_{0} is isomorphic to the algebra of virtual Laplace operators for the Lie algebra 𝔤𝔩(∞) [21]. Let λ = (λ_{1}, λ_{2}, …) be an integer sequence such that λ_{1} ≥ λ_{2} ≥ ⋯ and λ_{i} = 0 for all sufficiently large i and let L be a representation of 𝔤𝔩(∞) with the highest weight λ. Then every element a ∈ A_{0} acts in L as a scalar. A family of algebraically independent generators of A_{0} was constructed in [21, Remark 2.1.20] (see also [9]) and their eigenvalues in L were calculated. We shall need a different family of generators of A_{0} which will be more suitable for our purposes.

Let E = (E_{ij}) denote the infinite matrix whose ijth entry is the basis element E_{ij} of 𝔤𝔩(∞) and let E^{(n)} be its submatrix corresponding to the subset of indices 1 ≤ i, j ≤ n. It is well-known and can be easily verified that the mapping

T (u) ↦ 1 + E^{(n)}u^{−1}  (2.2)

defines an algebra epimorphism Y(n) → A(n). Consider the quantum determinant

qdet (1 + E^{(n)}u^{−1}) = ∑_{p∈𝔖_{n}} sgn(p) (1 + Eu^{−1})_{p(1)1} ⋯ (1 + E(u − n + 1)^{−1})_{p(n)n}

and write

qdet (1 + E^{(n)}u^{−1}) = 1 + E_{1}^{(n)}u^{−1} + E_{2}^{(n)}u^{−2} + · · · .

Since the coefficients of the series qdet T (u) are central in Y(n), the elements E_{i}^{(n)} belong to the center A_{0}(n) of A(n). Further, by definition (2.1) we obviously have

π_{n} : qdet (1 + E^{(n)}u^{−1}) ↦ qdet (1 + E^{(n−1)}u^{−1}).

So, we may define the elements E_{i} ∈ A_{0}, i = 1, 2, … as sequences E_{i} = (E_{i}^{(n)}|n ≥ 1). We call the corresponding generating series

qdet (1 + Eu^{−1}) := 1 + E_{1}u^{−1} + E_{2}u^{−2} + ⋯  (2.3)

the virtual quantum determinant. This is a particular case of a more general construction of quantum immanants given in [18, 19]. The following proposition is immediate.

Proposition 2.1 The elements E_{i} with i = 1, 2, … are algebraically independent generators of the algebra A_{0}. Their eigenvalues in a representation L of the Lie algebra 𝔤𝔩(∞) with the highest weight λ are given by the formula

qdet (1 + Eu^{−1})|_{L} = ∏_{i=1}^{∞} \frac{u + λ_{i} − i + 1}{u − i + 1}.

Consider the homomorphism Y(m) → Y(n) provided by Theorem 1.3 and take its composition with (2.2). We obtain an algebra homomorphism Φ_{n} : Y(m) → A(n) given by

Φ_{n} : t_{ij}(u) ↦ qdet (1 + Eu^{−1})_{B_{i}B_{j}},  (2.4)

where B_{i} denotes the set {i, m + 1, …, n}. In other words, Φ_{n}(t_{ij}(u)) coincides with the image of the element t̃_{ij}(u) under (2.2). Proposition 1.1 implies that this image commutes with the elements of the subalgebra 𝔤_{m}(n) and so, (2.4) defines a homomorphism Φ_{n} : Y(m) → A_{m}(n). Furthermore, the family of homomorphisms {Φ_{n}|n ≥ m} is obviously compatible with the homomorphisms (2.1) and thus defines an algebra homomorphism Φ : Y(m) → A_{m}. Introducing the virtual quantum determinants of the matrices (1 + Eu^{−1})_{B_{i}B_{j}} with B_{i} = {i, m + 1, m + 2, …} as in (2.3), we can represent this homomorphism as follows

Φ : t_{ij}(u) ↦ qdet (1 + Eu^{−1})_{B_{i}B_{j}}.

The following is a modified version of the Olshanski theorem; see [20, 21]. Denote by Ã_{0} the commutative algebra generated by the coefficients of the virtual quantum determinant qdet (1 + Eu^{−1})_{BB} with B = {m + 1, m + 2, …}.

Theorem 2.2 The homomorphism Φ : Y(m) → A_{m} is injective. Moreover, one has an isomorphism

A_{m} = Ã_{0} ⊗ Y(m),  (2.5)

where Y(m) is identified with its image under the embedding Φ.

Proof. Consider the canonical filtration of the universal enveloping algebra A(n). The corresponding graded algebra gr A(n) is isomorphic to the symmetric algebra P(n) of the space 𝔤𝔩(n). Elements of P(n) can be naturally identified with polynomials in matrix elements of an n × n-matrix x = (x_{ij}). Denote by P_{m}(n) the subalgebra of the elements of P(n) which are invariant under the adjoint action of the Lie algebra 𝔤_{m}(n). One can show (see [21]) that the graded algebra gr A_{m} is naturally isomorphic to the projective limit P_{m} of the commutative algebras P_{m}(n) with respect to homomorphisms P_{m}(n) → P_{m}(n − 1) analogous to (2.1). In particular, we can define the virtual determinants

det(1 + xu^{−1})_{B_{i}B_{j}},  det(1 + xu^{−1})_{BB}  (2.6)

of the submatrices of the infinite matrix 1 + xu^{−1}, where x = (x_{ij})_{i,j=1}^{∞}, as formal series in u^{−1} with coefficients from P_{m}. These coefficients coincide with the images in P_{m} of the coefficients of the virtual quantum determinants of the corresponding submatrices of E. Therefore, both claims of the theorem will follow from the fact that the coefficients of the series (2.6) where 1 ≤ i, j ≤ m are algebraically independent generators of the algebra P_{m}. To see this, for a finite value of n consider the sets B = {m + 1, m + 2, …, n} and B_{i} = {i, m + 1, m + 2, …, n}. We have an identity

det(1 + xu^{−1})_{B_{i}B_{j}} = det(1 + xu^{−1})_{BB} |(1 + xu^{−1})_{B_{i}B_{j}} |_{ij}

with

|(1 + xu^{−1})_{B_{i}B_{j}} |_{ij} = δ_{ij} + ∑_{k=1}^{∞}(−1)^{k−1}Λ_{ij}^{(k)}u^{−k},

where Λ_{ij}^{(k)} = ∑ x_{ia_{1}}x_{a_{1}a_{2}} ⋯ x_{a_{k−1}j}, summed over the indices a_{r} ∈ {m + 1, …, n}; see e.g. [7, Section 7.4]. However, the coefficients of the polynomial det(1 + xu^{−1})_{BB} generate the full set of invariants of the matrix x_{BB}. Further, these coefficients and the elements Λ_{ij}^{(k)} generate the algebra P_{m}(n) which follows from [21, Section 2.1.10] and [7, Section 7.4]. Finally, given any positive integer M, the coefficients at u^{−1}, …, u^{−M} of det(1+xu^{−1})_{BB} and the elements Λ_{ij}^{(k)} with k = 1, …, M and i, j = 1, …, m are algebraically independent for a sufficiently large n which is implied by [21, Lemma 2.1.11].

Remark. For m > 1 the algebra Ã_{0} in the decomposition (2.5) can be replaced by A_{0}. This can be deduced from the classical Sylvester’s identity (cf. Theorem 1.3):

det X = det(1 + xu^{−1}) (det(1 + xu^{−1})_{BB})^{m−1} ,

where X = (X_{ij}) with X_{ij} = det(1 + xu^{−1})_{B_{i}B_{j}}.

3 Extremal projection and transvector algebras

In the last two sections we shall consider the pair of Lie algebras 𝔤𝔩(m) ⊂ 𝔤𝔩(m + n) with the parameters m and n fixed, where 𝔤𝔩(m) is spanned by the basis elements E_{ij} of 𝔤𝔩(m + n) with i, j = 1, …, m.

Denote by 𝔥 the diagonal Cartan subalgebra of 𝔤𝔩(m) spanned by E_{ii} with i = 1, …, m. Consider the extension of the universal enveloping algebra A(m + n) = U(𝔤𝔩(m + n))

A′(m + n) = A(m + n) ⊗_{U(𝔥)} R(𝔥),

where R(𝔥) is the field of fractions of the commutative algebra U(𝔥). Let J denote the left ideal in A′(m + n) generated by the elements E_{ij} with 1 ≤ i < j ≤ m. The transvector algebra Z = Z(𝔤𝔩(m + n), 𝔤𝔩(m)) is the quotient algebra of the normalizer

Norm J = {x ∈ A′(m + n) | Jx ⊆ J}

modulo the two-sided ideal J; see [27]. It is an algebra over ℂ and an R(𝔥)-bimodule. Generators of Z can be constructed by using the extremal projection p for the Lie algebra 𝔤𝔩(m) [1, 26]. The projection p is an element of an algebra F of formal series and can be defined as follows. For any pair (i, j) such that 1 ≤ i < j ≤ m set

p_{ij} = ∑_{k=0}^{∞}(E_{ji})^{k}(E_{ij})^{k} \frac{(−1)^{k}}{k! (h_{i} − h_{j} + 1) ⋯ (h_{i} − h_{j} + k)},

where h_{i} := E_{ii} − i + 1. Then p is given by

p = ∏_{i<j} p_{ij},

where the product is taken in any normal ordering on the pairs (i, j). The positive roots of 𝔤𝔩(m) with respect to 𝔥 (with the standard choice of the positive root system) are naturally enumerated by the pairs (i, j) and normality means that any composite root lies between its components. The element p is, up to a factor from R(𝔥), a unique element of F such that

E_{ij} p = pE_{ji} = 0 for 1 ≤ i < j ≤ m.  (3.1)

In particular, p does not depend on the normal ordering and satisfies the conditions p^{2} = p and p^{∗} = p, where x ↦ x^{∗} is an involutive anti-automorphism of the algebra F such that (E_{ij})^{∗} = E_{ji}.

The action of p on elements of the quotient A′(m + n)/J is well-defined and the transvector algebra Z can be identified with the image of A′(m + n)/J with respect to the projection p:

Z = p (A′(m + n)/J) .

An analog of the Poincaré–Birkhoff–Witt theorem holds for the algebra Z [27, 28] so that ordered monomials in the elements E_{ab}, pE_{ia}, pE_{ai}, where i = 1, …, m and a, b = m + 1, …, m + n form a basis of Z as a left or right R(𝔥)-module. It will be convenient to use the following generators of Z: for i = 1, …, m and a = m + 1, …, m + n

s_{ia} = pE_{ia}(h_{i} − h_{1}) ⋯ (h_{i} − h_{i−1}),
s_{ai} = pE_{ai}(h_{i} − h_{i+1}) ⋯ (h_{i} − h_{m}).  (3.2)

Explicitly,

s_{ia} = ∑_{i>i_{1}>···>i_{s}≥1} E_{ii_{1}}E_{i_{1}i_{2}} ⋯ E_{i_{s−1}i_{s}}E_{i_{s}a}(h_{i} − h_{j_{1}}) ⋯ (h_{i} − h_{j_{r}}),  (3.3)
s_{ai} = ∑_{i<i_{1}<···<i_{s}≤m} E_{i_{1}i}E_{i_{2}i_{1}} ⋯ E_{i_{s}i_{s−1}}E_{ai_{s}}(h_{i} − h_{j_{1}}) ⋯ (h_{i} − h_{j_{r}}),

where s = 0, 1, … and {j_{1}, …, j_{r}} is the complementary subset to {i_{1}, …, i_{s}} respectively in the set {1, …, i − 1} or {i + 1, …, m}. The elements E_{ab}, a, b = m + 1, …, m + n generate a subalgebra in Z isomorphic to A(n). Moreover, one also has the following relations in Z which are obtained by using the methods of [27, 28]. Let the indices i, j and a, b, c run through the sets {1, …, m} and {m+1, …, m+n}, respectively. We have

[E_{ab}, s_{ci}] = δ_{bc}s_{ai},  [E_{ab}, s_{ic}] = −δ_{ac}s_{ib}.  (3.4)

Moreover, if i ≠ j then

s_{ai}s_{bj} = s_{bj}s_{ai} \frac{h_{i} − h_{j} + 1}{h_{i} − h_{j}} − s_{bi}s_{aj} \frac{1}{h_{i} − h_{j}} ,

while

s_{ai}s_{bi} = s_{bi}s_{ai}.  (3.5)

Finally, s_{ia}s_{bj} = s_{bj}s_{ia} for i ≠ j while

s_{ia}s_{bi} = (δ_{ab}(E_{ii}+m−i)−E_{ab}) ∏_{j=1, j≠i}^{m}(h_{i}−h_{j}−1)+∑_{j=1}^{m} s_{bj}s_{ja} ∏_{k=1, k≠j}^{m} \frac{h_{i} − h_{k} − 1}{h_{j} − h_{k}}. (3.6)

The anti-involution x ↦ x^{∗} extends to the algebra Z so that (pE_{ai})^{∗} = pE_{ia} [27, 28]. Therefore,

(s_{ai})^{∗} = s_{ia} \frac{(h_{i} − h_{i+1} + 1) ⋯ (h_{i} − h_{m} + 1)}{(h_{i} − h_{1}) ⋯ (h_{i} − h_{i−1})}.  (3.7)

The centralizer A = U(𝔤𝔩(m + n))^{𝔤𝔩(m)} is a subalgebra in the normalizer Norm J and so, one has the natural algebra homomorphism ν : A → Z. By the results of Section 2 we have an algebra homomorphism

ϕ : Y(n) → A,  t_{ab}(u) ↦ qdet (1 + Eu^{−1})_{C_{a}C_{b}},  (3.8)

where C_{a} = {1, …, m, m + a} for a = 1, …, n; see (2.4). The composition ψ = ν ∘ ϕ is an algebra homomorphism ψ : Y(n) → Z. The next theorem provides explicit formulas for the images of the generators of Y(n) with respect to ψ. Note that by (3.8) the image of

T_{m+a,m+b}(u) := u(u − 1) ⋯ (u − m) t_{ab}(u)  (3.9)

under ψ is a polynomial in u.

Theorem 3.1 We have for a, b = m + 1, …, m + n:

ψ : T_{ab}(u) ↦ (δ_{ab}(u − m) + E_{ab}) ∏_{i=1}^{m}(u + h_{i}) − ∑_{i=1}^{m} s_{ai}s_{ib} ∏_{j=1, j≠i}^{m} \frac{u + h_{j}}{h_{i} − h_{j}}  (3.10)

and

ψ : T_{ab}(u) ↦ (δ_{ab}u + E_{ab}) ∏_{i=1}^{m}(u + h_{i} − 1) − ∑_{i=1}^{m} s_{ib}s_{ai} ∏_{j=1, j≠i}^{m} \frac{u + h_{j} − 1}{h_{i} − h_{j}}. (3.11)

Proof. By (1.9) the image of T_{ab}(u) under the homomorphism (3.8) is

∑_{σ} sgn(σ) (u + E)_{σ(1),1} ⋯ (u + E − m + 1)_{σ(m),m}(u + E − m)_{σ(a),b},  (3.12)

summed over permutations σ of the set {1, …, m, a}. To find the image of the polynomial (3.12) in Z we shall regard it modulo the ideal J and apply the projection p. Consider first the sum in (3.12) over the permutations σ such that σ(a) = a. Using (3.1) we obtain that its image in Z[u] is

(u + h_{1}) ⋯ (u + h_{m})(δ_{ab}(u − m) + E_{ab}).  (3.13)

Further, consider the remaining summands in (3.12) and suppose that σ(a) = i, σ(j) = a for certain i, j ∈ {1, …, m}. Using the property (3.1) of the projection p we find that the image of such a summand in Z[u] can be nonzero only if i ≥ j and σ is a cyclic permutation of the form σ = (a, i_{1}, …, i_{s}) where i = i_{1} > i_{2} > ⋯ > i_{s} = j. We have the equality modulo J:

E_{ai_{s}}E_{i_{s}i_{s−1}} ⋯ E_{i_{2}i_{1}}E_{i_{1}b} = E_{aj}E_{jb}.

Since sgn(σ) = (−1)^{s} the image in Z[u] of the summand corresponding to σ is

(−1)^{s} pE_{aj}E_{jb}(u + h_{1}) ⋯ { \atop (u + h_{i_{s}})} ⋯ { \atop (u + h_{i_{1}})} ⋯ (u + h_{m}),

where the hats indicate the factors to be omitted. Taking the sum over cycles σ we find that the polynomial S_{ab}(u) := ψ(T_{ab}(u)) is given by

S_{ab}(u) = (δ_{ab}(u − m) + E_{ab})(u + h_{1}) ⋯ (u + h_{m})
− ∑_{j=1}^{m} pE_{aj}E_{jb}(u + h_{1}) ⋯ (u + h_{j−1})(u + h_{j+1} − 1) ⋯ (u + h_{m} − 1).

Consider the value of S_{ab}(u) at u = −h_{i} for i ∈ {1, …, m} (to make this evaluation well-defined we agree to write the coefficients of the polynomial to the left of the powers of u). We obtain

S_{ab}(−h_{i}) = − ∑_{j=1}^{i} pE_{aj}E_{jb}(h_{1} − h_{i}) ⋯ (h_{j−1} − h_{i})(h_{j+1} − h_{i} − 1) ⋯ (h_{m} − h_{i} − 1).

On the other hand, by (3.2) we have

s_{ai}s_{ib} = pE_{ai} s_{ib}(h_{i} − h_{i+1} + 1) ⋯ (h_{i} − h_{m} + 1).

Finally, using (3.3) and (3.1) we verify the identity s_{ai}s_{ib} = (−1)^{m}S_{ab}(−h_{i}). The difference of S_{ab}(u) and the product (3.13) is a polynomial in u of degree < m. So, (3.10) follows from the Lagrange interpolation formula.

The relation (3.11) follows from (3.6) and (3.10); it suffices to consider the values of the polynomial S_{ab}(u) at u = −h_{i} +1 for i = 1, …, m. It can also be proved by the above argument where (1.9) is applied to t _{b,1··· m}^{a,1··· m}(u) (thus giving another derivation of (3.6)).

4 Elementary representations of the Yangian

Here we use the results of the previous section to study a special class of representations of the Yangian Y(n) called elementary. They play an important role in the classification of the representations of Y(n) with a semisimple action of the Gelfand– Tsetlin subalgebra; see [17].

4.1 Yangian action on the multiplicity space

A representation L of the Yangian Y(n) is called highest weight if it is generated by a nonzero vector ζ (the highest vector) such that

t_{aa}(u)ζ = λ_{a}(u)ζ  for a = 1, …, n,
t_{ab}(u)ζ = 0  for 1 ≤ a < b ≤ n

for certain formal series λ_{a}(u) ∈ 1 + u^{−1}ℂ[[u^{−1}]]. The set λ(u) = (λ_{1}(u), …, λ_{n}(u)) is called the highest weight of L; cf. [2, 6]. Every finite-dimensional irreducible representation of the Yangian Y(n) is highest weight. It contains a unique, up to scalar multiples, highest vector. An irreducible representation of Y(n) with the highest weight λ(u) is finite-dimensional if and only if there exist monic polynomials P_{1}(u), …, P_{n−1}(u) in u (called the Drinfeld polynomials) such that

\frac{λ_{a}(u)}{λ_{a+1}(u)} = \frac{P_{a}(u + 1)}{P_{a}(u)},  a = 1, …, n − 1.  (4.1)

These results are contained in [6]; see also [2, 13].

Let λ = (λ_{1}, …, λ_{m+n}) be an (m + n)-tuple of complex numbers satisfying the condition λ_{i} − λ_{i+1} ∈ ℤ_{+} for i = 1, …, m + n − 1. Denote by L(λ) the irreducible finite-dimensional representation of the Lie algebra 𝔤𝔩(m+n) with the highest weight λ. It contains a unique nonzero vector ξ (the highest vector) such that

E_{ii}ξ = λ_{i}ξ  for i = 1, …, m + n,
E_{ij}ξ = 0  for 1 ≤ i < j ≤ m + n.

Given a 𝔤𝔩(m)-highest weight μ = (μ_{1}, …, μ_{m}) we denote by L(λ)_{μ}^{+} the subspace of 𝔤𝔩(m)-highest vectors in L(λ) of weight μ:

L(λ)_{μ}^{+} = {η ∈ L(λ) | E_{ii}η = μ_{i}η for i = 1, …, m,
E_{ij}η = 0  for 1 ≤ i < j ≤ m}.

The dimension of L(λ)_{μ}^{+} coincides with the multiplicity of the 𝔤𝔩(m)-module L(μ) in the restriction of L(λ) to 𝔤𝔩(m). The multiplicity space L(λ)_{μ}^{+} admits a natural structure of an irreducible representation of the centralizer algebra A [4, Section 9.1]. On the other hand, the homomorphism (3.8) allows us to regard L(λ)_{μ}^{+} as a Y(n)- module. As it follows from the proof of Theorem 2.2, the algebra A is generated by the image of ϕ and the center of U(𝔤𝔩(m)). The elements of the center act in L(λ)_{μ}^{+} as scalar operators and so, the Y(n)-module L(λ)_{μ}^{+} is irreducible. Following [17] we call it elementary.

We shall use a contravariant bilinear form ⟨ , ⟩ on the space L(λ) which is uniquely determined by the conditions

⟨ξ, ξ⟩ = 1; ⟨Xu, v⟩ = ⟨u, X^{∗}v⟩ for u, v ∈ L(λ), X ∈ A(m + n),

where X ↦ X^{∗} is the anti-involution on A(m + n) such that (E_{ab})^{∗} = E_{ba}. The form ⟨ , ⟩ is nondegenerate on L(λ), and so is its restriction to each nonzero subspace L(λ)_{μ}^{+}. Indeed, if there is a vector η ∈ L(λ)_{μ}^{+} such that ⟨η, η′⟩ = 0 for each η′ ∈ L(λ)_{μ}^{+} then η is also orthogonal to all elements of L(λ)^{+} because distinct 𝔤𝔩(m)- weight subspaces in L(λ)^{+} are pairwise orthogonal to each other. On the other hand, L(λ) = U(𝔫_{−}) L(λ)^{+}, where 𝔫_{−} is the lower triangular subalgebra in 𝔤𝔩(m). Therefore, η is orthogonal to L(λ) since for any X ∈ 𝔫_{−} and η′ ∈ L(λ) one has

⟨η, Xη′⟩ = ⟨X^{∗}η, η′⟩ = 0

because X^{∗}η = 0. Thus, η = 0.

We shall maintain the notation ⟨ , ⟩ for the restriction of the form to a subspace L(λ)_{μ}^{+}.

4.2 Quantum minor formulas for generators of Z

Using the homomorphism Y(m + n) → A(m + n) defined by (2.2) we can carry over the quantum minor identities from Section 1 to the matrix 1 + Eu^{−1} or, which is more convenient, to E(u) = u + E. In accordance with (1.9) and (1.10) we define the polynomials E _{b_{1}··· b_{s}}^{a_{1}··· a_{s}}(u) by the following equivalent formulas

E _{b_{1}··· b_{s}}^{a_{1}··· a_{s}}(u) = ∑_{σ∈𝔖_{s}} sgn(σ) (u + E)_{a_{σ(1)}b_{1}} ⋯ (u + E − s + 1)_{a_{σ(s)}b_{s}}  (4.2)
= ∑_{σ∈𝔖_{s}} sgn(σ) (u + E − s + 1)_{a_{1}b_{σ(1)}} ⋯ (u + E)_{a_{s}b_{σ(s)}}.  (4.3)

The polynomial E _{b_{1}··· b_{s}}^{a_{1}··· a_{s}}(u) is skew-symmetric with respect to permutations of the upper indices and of the lower indices. We shall need to evaluate u in R(𝔥). To make this operation well-defined let us agree to write the coefficients of this polynomial to the left from powers of u. In the following proposition we use the notation from Section 3.

Proposition 4.1 For any i ∈ {1, …, m} and a ∈ {m + 1, …, m + n} we have the equalities in A′(m + n) modulo the ideal J:

s_{ia} = (−1)^{i−1}E _{1··· i−1,a}^{1··· i}(−h_{i}),  (4.4)
s_{ai} = E _{i··· m}^{i+1··· m,a}(−h_{i} − i + 1).  (4.5)

Proof. Let us show first that E _{i··· m}^{i+1··· m,a}(−E_{ii}) (recall that h_{i} = E_{ii} − i + 1) belongs to the normalizer Norm J of J. By Proposition 1.1 we have the relations

[E_{ij}, E _{b_{1}··· b_{s}}^{a_{1}··· a_{s}}(u)] = ∑_{r=1}^{s} (δ_{ja_{r}}E _{b_{1}··· b_{s}}^{a_{1}··· i··· a_{s}}(u) − δ_{ib_{r}}E _{b_{1}··· j··· b_{s}}^{a_{1}··· a_{s}}(u)) ,

where i and j on the right hand side take the rth positions. This implies that E_{k,k+1} commutes with E _{i··· m}^{i+1··· m,a}(u) for k ∈ {1, …, m − 1}, k ≠ i. Hence,

E_{k,k+1}E _{i··· m}^{i+1··· m,a}(−E_{ii}) ∈ J.

Furthermore,

[E_{i,i+1}, E _{i··· m}^{i+1··· m,a}(u)] = E _{i··· m}^{i,i+2··· m,a}(u).

We have

E _{i··· m}^{i,i+2··· m,a}(u) = (−1)^{m−i}E _{i··· m}^{i+2··· m,a,i}(u),

and by (4.3)

E _{i··· m}^{i+2··· m,a,i}(u) = ∑_{σ} sgn(σ) (u + E − m + i)_{i+2,σ(i)} ⋯ (u + E)_{i,σ(m)}. (4.6)

However, σ(m) take values in {i, …, m} and so, the element (4.6) belongs to the ideal J for u = −E_{ii}. Note that E_{ii} belongs to the normalizer Norm J and therefore E _{i··· m}^{i+1··· m,a}(−E_{ii}) does.

To complete the proof of (4.5) we calculate the image E _{i··· m}^{i+1··· m,a}(−E_{ii}) under the extremal projection p. By (4.2) we have

E _{i··· m}^{i+1··· m,a}(u) = ∑_{σ} sgn(σ) (u + E)_{σ(i+1),i} ⋯ (u + E − m + i)_{σ(a),m}. (4.7)

By the property (3.1) of p we obtain that the image of a summand in (4.7) under p is zero unless σ(i + 1) = a. Further, a summand is zero modulo the ideal J unless

σ(a) = m, σ(m) = m − 1, …, σ(i + 2) = i + 1.

Therefore, the image of (4.7) with u = −E_{ii} equals

pE_{ai} (h_{i} − h_{i+1}) ⋯ (h_{i} − h_{m}),

which coincides with s_{ai}. This proves (4.5). The relation (4.4) is proved by a similar argument.

4.3 Highest vector of the Y(n)-module L(λ)_{μ}^{+}

Our aim now is to explicitly construct the highest vector of the Y(n)-module L(λ)_{μ}^{+} and to find its highest weight. This will also allow us to calculate its Drinfeld polynomials.

The action of the Yangian Y(n) in L(λ)_{μ}^{+} is determined by Theorem 3.1. From now on we shall assume that the highest weight λ is a partition, that is, the λ_{i} are nonnegative integers. This does not lead to a real loss of generality because the formulas and arguments below can be easily adjusted to be valid in the general case. Given a general λ one can add a suitable complex number to all entries of λ to get a partition.

As it follows from the branching rule for the general linear Lie algebras (see [25]) the space L(λ)_{μ}^{+} is nonzero only if μ is a partition such that μ ⊂ λ and each column of the skew diagram λ/μ contains at most n cells. The figure below illustrates the skew diagram for λ = (10, 8, 5, 4, 2) and μ = (6, 3).

Introduce the row order on the cells of λ/μ corresponding to reading the diagram by rows from left to right starting from the top row. For a cell α ∈ λ/μ denote by r(α) the row number of α and by l(α) the (increased) leglength of α which equals 1 plus the number of cells of λ/μ in the column containing α which are below α. Consider the following element of L(λ):

ζ = ∏_{α∈λ/μ, r(α)≤m} s_{m+l(α),r(α)} ξ,  (4.8)

where ξ is the highest vector of L(λ) and the product is taken in the row order. Using (3.4) we find that ζ ∈ L(λ)_{μ}^{+}. For the above example of λ/μ we have m = 2, n = 3 and

ζ = (s_{41})^{2} (s_{31})^{2} s_{52} s_{42} (s_{32})^{3} ξ.

Proposition 4.1 allows us to rewrite the definition (4.8) in a different form. Introduce the notation

τ_{ai}(u) = E _{i··· m}^{i+1··· m,a}(u), τ_{ia}(u) = E _{1··· i−1,a}^{1··· i}(u).

Then by (4.5) we have

ζ = ∏_{α∈λ/μ, r(α)≤m} τ_{m+l(α),r(α)}(−c(α)) ξ,

where c(α) is the column number of α, and the product is taken in the row order.

Given three integers i, j, k we shall denote by middle{i, j, k} that of the three which is between the two others.

Theorem 4.2 The vector ζ defined by (4.8) is the highest vector of the Y(n)-module L(λ)_{μ}^{+}. The highest weight of this module is (λ_{1}(u), …, λ_{n}(u)) where

λ_{a}(u) = \frac{(u + ν_{a}^{(1)})(u + ν_{a}^{(2)} − 1) ⋯ (u + ν_{a}^{(m+1)} − m)}{u(u − 1) ⋯ (u − m)}  (4.9)

and

ν_{a}^{(i)} = middle{μ_{i−1}, μ_{i}, λ_{a+i−1}}  (4.10)

with μ_{m+1} = 0, and μ_{0} is considered to be sufficiently large.

Note that for each i the n-tuple ν^{(i)} = (ν_{1}^{(i)}, …, ν_{n}^{(i)}) is a partition which can be obtained from λ/μ as follows. Consider the subdiagram of λ of the form λ^{(i)} = (λ_{i}, λ_{i+1}, …, λ_{i+n−1}). Replace the rows of λ^{(i)} which are longer than μ_{i−1} by μ_{i−1} while those which are shorter than μ_{i} replace with μ_{i} and leave the remaining rows unchanged. The resulting partition is ν^{(i)}. For the above example with λ = (10, 8, 5, 4, 2) and μ = (6, 3) we have

ν^{(1)} = (10, 8, 6),  ν^{(2)} = (6, 5, 4),  ν^{(3)} = (3, 3, 2),

as illustrated:

Proof of Theorem 4.2. Introduce two parameters k and l of the diagram λ/μ as follows. We let k be the row number of the top (non-empty) row of λ/μ if this number is less or equal to m; otherwise set k = m + 1. So, if k ≤ m then we have μ_{i} = λ_{i} for i = 1, …, k − 1 and μ_{k} < λ_{k}. We denote by l the leglength of the cell α = (k, μ_{k} + 1) with k ≤ m. We shall prove the following three relations simultaneously by induction on k and l (see (3.9) for the definition of T_{ab}(u)):

T_{ab}(u) ζ = 0  (4.11)

for m + 1 ≤ a < b ≤ m + n,

T_{m+a,m+a}(u) ζ = (u + ν_{a}^{(1)})(u + ν_{a}^{(2)} − 1) ⋯ (u + ν_{a}^{(m+1)} − m) ζ  (4.12)

for a = 1, …, n, and

s_{m+l,k} ζ = 0,  (4.13)

if k ≤ m and μ_{k} = λ_{k+l}.

Suppose that k = m + 1. Then ζ = ξ and both (4.11) and (4.12) are immediate from (3.10). Now let k ≤ m be fixed. We assume that (4.11)–(4.13) hold for all greater values of the parameter k. We proceed by induction on l.

To simplify the notation we shall identify the series T_{ab}(u) with its image under the homomorphism (3.8). That is, we set

T_{ab}(u) = E _{1··· m,b}^{1··· m,a}(u).

We have

T_{ab}(u)^{∗} = T_{ba}(u),  (4.14)

which follows from (4.2) and (4.3). The following relations are derived from Proposition 1.1:

T_{ab}(u)τ_{ci}(v) = τ_{ci}(v)T_{ab}(u) \frac{u − v − i}{u − v − i + 1} + τ_{ai}(v)T_{cb}(u) \frac{1}{u − v − i + 1}, (4.15)
T_{ab}(u)τ_{ci}(v) = τ_{ci}(v)T_{ab}(u) \frac{u − v − i − 1}{u − v − i} + T_{cb}(u)τ_{ai}(v) \frac{1}{u − v − i}.  (4.16)

Using (4.5) replace the first factor s_{m+l,k} in (4.8) by τ_{m+l,k}(−μ_{k} − 1) so that the vector ζ will be written in the form ζ = τ_{m+l,k}(−μ_{k} − 1)ζ′. Apply the operator T_{ab}(u) with a < b to ζ. If b > m + l then by (4.15) and (4.5) we have

T_{ab}(u)ζ = \frac{u + μ_{k} − k + 1}{u + μ_{k} − k + 2} s_{m+l,k}T_{ab}(u) ζ′ + \frac{1}{u + μ_{k} − k + 2} s_{ak}T_{m+l,b}(u) ζ′. (4.17)

Now (4.11) follows by induction on l and the degree of ζ with respect to s_{m+l,k}. If b ≤ m + l then a < m + l and by (4.16) and (4.5) we have

T_{ab}(u)ζ = \frac{u + μ_{k} − k}{u + μ_{k} − k + 1} s_{m+l,k}T_{ab}(u) ζ′ + \frac{1}{u + μ_{k} − k + 1} T_{m+l,b}(u)s_{ak} ζ′. (4.18)

However, by (3.5) we have s_{ak} ζ′ = 0, where we have used the induction hypothesis for (4.13). So, (4.11) follows again by an obvious induction.

Now we use a similar argument to calculate T_{aa}(u) ζ. The relation (4.17) with a = b gives

T_{aa}(u)ζ = \frac{u + μ_{k} − k + 1}{u + μ_{k} − k + 2} s_{m+l,k}T_{aa}(u) ζ′

for a > m + l, and

T_{aa}(u)ζ = s_{m+l,k}T_{aa}(u) ζ′

for a = m + l. If a < m + l then (4.18) with a = b gives

T_{aa}(u)ζ = \frac{u + μ_{k} − k}{u + μ_{k} − k + 1} s_{m+l,k}T_{aa}(u) ζ′.

In all the cases the relation (4.12) follows by an obvious induction.

Let us now prove (4.13). Suppose that μ_{k} = λ_{k+l} but ζ̃:= s_{m+l,k} ζ ≠ 0. Then ζ̃ is a 𝔤𝔩(m)-highest vector of weight μ̃ = μ − δ_{k}. This is only possible if μ_{k} > μ_{k+1} which will be assumed. We can repeat the previous arguments where the vector ζ is replaced with ζ̃ to show that ζ̃ is annihilated by T_{ab}(u) with a < b and ζ̃ is an eigenvector for the T_{aa}(u). This means that ζ̃ is the highest vector of the Y(n)-module L(λ)+ ; μ̃. Let us verify that this vector is orthogonal to all elements of L(λ)+ ; μ̃. Indeed, by the Poincaré–Birkhoff–Witt theorem for the algebra Y(n) (see e.g. [14, Corollary 1.23]), L(λ)+ ; μ̃ = Y_{−} ζ̃, where Y_{−} is the span of monomials in the coefficients of the series T_{ba}(u) with b > a. However, due to (4.14) we have for any η ∈ L(λ)+ ; μ̃:

⟨ζ̃, T_{ba}(u)η⟩ = ⟨T_{ab}(u)ζ̃, η⟩ = 0.

It remains to check that ⟨ζ̃, ζ̃⟩ = 0. Indeed, by (3.7)

⟨ζ̃, ζ̃⟩ = const · ⟨ζ, s_{k,m+l}s_{m+l,k}ζ⟩.

By (3.11) we have

s_{k,m+l}s_{m+l,k}ζ = (−1)^{m}T_{m+l,m+l}(−h_{k} + 1) ζ.  (4.19)

Note that

h_{k} ζ = (μ_{k} − k + 1) ζ = (λ_{k+l} − k + 1) ζ.

Therefore, (4.19) equals (−1)^{m}T_{m+l,m+l}(−λ_{k+l} + k) ζ which is zero by (4.12). Thus, ζ̃ has to be zero which contradicts to our assumption.

Finally, to show that ζ ≠ 0 we apply appropriate operators s_{ia} to ζ repeatedly to get the highest vector ξ of L(λ) with a nonzero coefficient. Indeed, by (3.11) we have

s_{k,m+l} ζ = (−1)^{m}T_{m+l,m+l}(−μ_{k} + k − 1) ζ′

which equals

−(μ_{k} − ν_{l}^{(1)} − k + 1) ⋯ (μ_{k} − ν_{l}^{(m+1)} − k + m + 1) ζ′  (4.20)

by (4.12). Here the numbers ν_{l}^{(i)} are defined by (4.10) with

μ = (λ_{1}, …, λ_{k−1}, μ_{k} + 1, μ_{k+1}, …, μ_{m}).

So, the coefficient at ζ′ in (4.20) is nonzero.

Theorem 4.2 implies the following corollary. As usual, by the content of a cell α = (i, j) ∈ λ/μ we mean the number j − i.

Corollary 4.3 The Drinfeld polynomials for the Y(n)-module L(λ)_{μ}^{+} are given by

P_{a}(u) = ∏_{c}(u + c),  a = 1, …, n − 1,  (4.21)

where c runs over the contents of the top cells of columns of height a in the diagram λ/μ.

Proof. By the definition (4.1) and the formula (4.9) we have

P_{a}(u) = ∏_{k=1}^{m+1}(u + ν_{a+1}^{(k)} − k + 1) ⋯ (u + ν_{a}^{(k)} − k),

where the kth factor is assumed to be equal to 1 if ν_{a}^{(k)} = ν_{a+1}^{(k)}. Now (4.10) implies that this product coincides with (4.21).

If λ = (10, 8, 5, 4, 2) and μ = (6, 3) (see the example above) then we have

P_{1}(u) = (u + 4)(u + 8)(u + 9), P_{2}(u) = u(u + 3)(u + 6)(u + 7).

The corollary shows that the Y(𝔰𝔩(n))-module L(λ)_{μ}^{+} is isomorphic to the corresponding elementary representation of Y(𝔰𝔩(n)) which was studied in [17].

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