Small Elliptic Quantum Group e_{τ,γ}(𝔰𝔩_{N})

V.Tarasov^{⋆} and A.Varchenko^{∗}

Abstract. The small elliptic quantum group e_{τ,γ}(𝔰𝔩_{N}) , introduced in the paper, is an elliptic dynamical analogue of the universal enveloping algebra U(𝔰𝔩_{N}) . We define highest weight modules, Verma modules and contragradient modules over e_{τ,γ}(𝔰𝔩_{N}) , the dynamical Shapovalov form for e_{τ,γ}(𝔰𝔩_{N}) and the contravariant form for highest weight e_{τ,γ}(𝔰𝔩_{N})-modules. We show that any finite-dimensional 𝔰𝔩_{N}-module and any Verma module over 𝔰𝔩_{N} can be lifted to the corresponding e_{τ,γ}(𝔰𝔩_{N})-module on the same vector space. For the elliptic quantum group E_{τ,γ}(𝔰𝔩_{N}) we construct the evaluation morphism E_{τ,γ}(𝔰𝔩_{N}) → e_{τ,γ}(𝔰𝔩_{N}) , thus making any e_{τ,γ}(𝔰𝔩_{N})-module into an evaluation E_{τ,γ}(𝔰𝔩_{N})-module.

Introduction

The main purpose of this paper is to define a dynamical quantum group e_{τ,γ}(𝔰𝔩_{N}) which is an elliptic dynamical analogue of the universal enveloping algebra U(𝔰𝔩_{N}) . We call e_{τ,γ}(𝔰𝔩_{N}) the small elliptic quantum group, comparing it with the elliptic quantum group E_{τ,γ}(𝔰𝔩_{N}) introduced in [F]. Our initial motivation to study this object arises from the wish to understand the structure of evaluation modules over E_{τ,γ}(𝔰𝔩_{N}) , which should be analogous to evaluation modules over the Yangian Y (𝔰𝔩_{N}) .

Evaluation modules over E_{τ,γ}(𝔰𝔩_{2}) have been defined in [FV1]. They appear naturally in the descripion of transition matrices for the trigonometric qKZ difference equation [TV1]. They also serve for the definiton of the qKZB difference equations and occur in the description of its monodromies, see [FTV]. One should expect evaluation modules over E_{τ,γ}(𝔰𝔩_{N}) for N > 2 to play a similar role. Symmetric and exterior powers of the vector representation of E_{τ,γ}(𝔰𝔩_{N}) , developed in [FV2], are examples of evaluation modules over E_{τ,γ}(𝔰𝔩_{N}) . In general, evaluation modules over E_{τ,γ}(𝔰𝔩_{N}) arise from e_{τ,γ}(𝔰𝔩_{N})-modules via the evaluation morphism E_{τ,γ}(𝔰𝔩_{N}) → e_{τ,γ}(𝔰𝔩_{N}) , see Corollary 3.4, analogous to the evaluation homomorphism Y (𝔰𝔩_{N}) → U(𝔰𝔩_{N}) .

In this paper we prove the PBW type theorem for the small elliptic quantum group e_{τ,γ}(𝔰𝔩_{N}) . We define highest weight modules, Verma modules and contragradient modules over e_{τ,γ}(𝔰𝔩_{N}) . We show that for any finite-dimensional 𝔰𝔩_{N}-module and any Verma module over 𝔰𝔩_{N} one can define the corresponding e_{τ,γ}(𝔰𝔩_{N})-module on the same vector space. Pulling them back through the evaluation morphism we get finite-dimensional evaluation modules and evaluation Verma modules over E_{τ,γ}(𝔰𝔩_{N}) . Conjecturally, the same picture takes place for any highest weight 𝔰𝔩_{N}-module.

We introduce the dynamical Shapovalov form for e_{τ,γ}(𝔰𝔩_{N}) , the dynamical Shapovalov pairing and the contravariant form for the highest weight modules over e_{τ,γ}(𝔰𝔩_{N}) . They play an important role in the construction of finite-dimensional highest weight e_{τ,γ}(𝔰𝔩_{N})-modules. From another point of view the contravariant form for e_{τ,γ}(𝔰𝔩_{2})-modules appeared in the disguised form in [TV, Appendix C].

The small elliptic quantum group e_{τ,γ}(𝔰𝔩_{N}) admits the trigonometric and rational degenerations. They are closely related to the exchange quantum groups F_{q}(SL(N)) and F (SL(N)) introduced in [EV2]. In this paper we consider only the rational dynamical quantum group e_{rat}(𝔰𝔩_{N}) and its relation to the exchange quantum group F (SL(N)). We contstruct a functor from a certain category of 𝔰𝔩_{N}-modules to a category of e_{rat}(𝔰𝔩_{N})-modules, see Theorems 7.5, 7.6, and 9.9. We also establish an equivalence of certain tensor categories of e_{rat}(𝔰𝔩_{N})-modules and rational dynamical representations of F (SL(N)), see Theorem 10.4. In particular, this gives a new contruction of highest weight representations for the exchange quantum group F (SL(N)).

Notice that while e_{rat}(𝔰𝔩_{N}) and exchange quantum groups have a coproduct structure, no coproduct structure is known for the elliptic quantum group e_{τ,γ}(𝔰𝔩_{N}) . This makes the small elliptic group similar to the Sklyanin algebra [S], [HW].

⋆Supported in part by RFFI grant 99–01–00101 and INTAS grant 99–01705

E-mail: tarasov@mpim-bonn.mpg.de , vt@pdmi.ras.ru

∗Supported in part by NSF grant DMS–9801582

E-mail: av@math.unc.edu

The paper is organized as follows. After introducing basic notation we recall the definiton of the elliptic quantum group E_{τ,γ}(𝔰𝔩_{N}) . The small elliptic quantum group e_{τ,γ}(𝔰𝔩_{N}) is defined in Section 3. In Section 4 we introduce highest weight modules and Verma modules over e_{τ,γ}(𝔰𝔩_{N}) . The dynamical Shapovalov form for e_{τ,γ}(𝔰𝔩_{N}) is defined in Section 5. Contragradient modules and the contravariant form for highest weight e_{τ,γ}(𝔰𝔩_{N})-modules are defined in Section 6. In Section 7 we study the rational dynamical quantum group e_{rat}(𝔰𝔩_{N}) . We construct irreducible finite-dimensional e_{τ,γ}(𝔰𝔩_{N})-modules in Section 8. A functor from a certain category of 𝔰𝔩_{N}-modules to a category of e_{rat}(𝔰𝔩_{N})-modules is defined in Section 9. Relations between e_{rat}(𝔰𝔩_{N}) and F (SL(N)) are studied in Section 10. There are six Appendices in the paper; they contain useful technichal information, the example of e_{τ,γ}(𝔰𝔩_{2}) , and some proofs.

The first author would like to thank for hospitality the Max-Planck-Institut für Mathematik in Bonn, where he stayed when this paper was being written.

1. Basic notation

Let τ be a complex number such that Im τ > 0 . Let θ(u; τ) be the Jacobi theta function:

θ(u; τ) = − ∑_{m=−∞}^{∞}exp(πiτ (m + 1/2)^{2} + 2πi(m + 1/2)(u + 1/2)) .

There is a product formula

θ(u; τ) = 2e^{πiτ/4} sin(πu) ∏_{s=1}^{∞}(1 − e^{2πisτ})(1 − e^{2πi(sτ+u)})(1 − e^{2πi(sτ−u)}) .

The function θ(u; τ) has multipliers −1 and − exp(−2πiu − πiτ) as u → u + 1 and u → u + τ , respectively. θ(u; τ) is an entire function with only simple zeros lying on the lattice ℤ +τ ℤ. Usually, we do not indicate the second argument of the theta function explicitly, writing θ(u) instead of θ(u; τ) .

Let 𝔥 be a finite-dimensional commutative Lie algebra and let 𝔥^{∗} be the dual space. An 𝔥-module V is called diagonalizable if it admits a weight decomposition

V = ⨁_{μ ∈ 𝔥^{∗}}V [μ] ,

all weight subspaces V [μ] are finite-dimensional and the set {μ | V [μ] ≠ 0} is at most countable.

Let V_{1}, …, V_{k} be 𝔥-modules. For any function f : 𝔥^{∗} → End(V_{1} ⊗ … ⊗ V_{k}) and any i = 1, …, k we define an operator f(h^{(i)}) ∈ End(V_{1} ⊗ … ⊗ V_{k}) by the rule:

(1.1)  f(h^{(i)})v_{1} ⊗ … ⊗ v_{N} = f(μ)v_{1} ⊗ … ⊗ v_{N}  for any v ∈ V_{1} ⊗ … ⊗ V_{i}[μ] ⊗ … ⊗ V_{k} .

For a finite-dimensional vector space V over ℂ denote by Fun(V) the space of V-valued meromorphic functions on 𝔥^{∗}. If V is a diagonalizable 𝔥-module, set

Fun(V) = ⨁_{μ ∈ 𝔥^{∗}}Fun(V [μ]) .

Fun(V) is a vector space over Fun(ℂ) . The space V is naturally embedded in Fun(V) as the subspace of constant functions. If V is an 𝔥-module, then Fun(V) is an 𝔥-module with the natural pointwise action of 𝔥 and Fun(V)[μ] = Fun(V [μ]).

Let U be a diagonalizable 𝔥-module and a vector space over Fun(ℂ) . Suppose that the action of 𝔥 commutes with multiplication by functions. Then each weight subspace U[μ] is a vector space over Fun(ℂ) . Assume that all the weight subspaces are finite-dimensional over Fun(ℂ) . Then one can define a diagonalizable 𝔥-module V , such that U = Fun(V) as 𝔥-modules, in the following way. For any μ

such that U[μ] ≠ 0 pick up a basis f_{1}, …, f_{k} of U[μ] over Fun(ℂ) and set V [μ] = ⨁_{i=1}^{k} ℂf_{i}, otherwise,

set V [μ] = 0 . Then define V = ⨁_{μ ∈ 𝔥^{∗}}V [μ] to be the diagonalizable 𝔥-module such that V [μ] is a weight subspace of weight μ.

Let V, W be diagonalizable 𝔥-modules. The space Hom(V, W) has the natural 𝔥-module structure, but the weight subspaces are infinite-dimensional in general. We set

Fun(Hom(V, W)) = Hom(V, Fun(W)) .

A function φ ∈ Fun(Hom(V, W)) induces a linear map Fun(V) → Fun(W) , acting pointwise: f(λ) ↦ φ(λ)f(λ) . This map is usually denoted by the same letter.

Denote by D(V) the space of difference operators acting in Fun(V) . It is spanned over ℂ by operators of the form f(λ) ↦ φ(λ)f(λ + μ) where φ ∈ Fun(End(V)) and μ ∈ 𝔥^{∗}.

As a rule we do not distinguish a function φ(λ) ∈ Fun(ℂ) and a function φ(λ)·id ∈ Fun(End(V)).

In this paper we take 𝔥 to be the Cartan subalgebra of the Lie algebra 𝔰𝔩_{N} . Fix a basis h_{1}, …, h_{N−1} of 𝔥. Let ω_{1}, …, ω_{N−1} be the fundamental weights: ⟨ω_{a} , h_{b}⟩ = δ_{ab}. Let ℙ ⊂ 𝔥^{∗} be the weight lattice:

ℙ = ⨁_{a=1}^{N−1} ℤ ω_{a}. For any a = 1, …, N set ε_{a} = ω_{a} − ω_{a−1} , where by convention ω_{0} = ω_{N} = 0 . Let α_{1},

…, α_{N−1} be the simple roots: α_{a} = ε_{a} − ε_{a+1} . For λ, μ ∈ 𝔥^{∗} say that λ ⩾ μ if λ − μ ∈ ⨁_{a=1}^{N−1} ℤ_{⩾0} α_{a}.

Define a bilinear form ( ,) on 𝔥^{∗} by the rule (α_{a}, ω_{b}) = δ_{ab} for any a, b = 1, …, N − 1 . For any

λ ∈ 𝔥^{∗} set λ_{a} = (λ, ε_{a}) . It is easy to see that λ = ∑_{a=1}^{N}λ_{a} ε_{a}, ∑_{a=1}^{N}λ_{a} = 0 and (λ, μ) = ∑_{a=1}^{N}λ_{a}μ_{a}. The

Weyl group W acts on 𝔥^{∗} as the symmetric group S_{N} permuting the coordinates λ_{1}, …, λ_{N} .

Let ρ = ∑_{a=1}^{N−1}ω_{a} = − ∑_{a=1}^{N}aε_{a} be the half-sum of positive roots. For any w ∈ W and λ ∈ 𝔥^{∗} set

w ·λ = w(λ + ρ) − ρ. Notice that (λ, ρ) = − λ_{1} − 2λ_{2} − … − Nλ_{N} .

Let E_{ab} ∈ End(ℂ^{N}) be the matrix with the only nonzero entry equal to 1 at the intersection of the a-th row and b-th column. The assignment h_{a} ↦ E_{aa} − E_{a+1,a+1} , a = 1, …, N − 1 , makes ℂ^{N} into an 𝔥-module, called the vector representation of 𝔥. Henceforth, we always consider ℂ^{N} as the vector representation of 𝔥.

Let γ be a nonzero complex number. Introduce functions α(u, ξ) and β(u, ξ) as follows:

(1.2)  α(u, ξ) = \frac{θ(u)θ(ξ + γ)}{θ(u − γ)θ(ξ)} ,  β(u, ξ) = − \frac{θ(u + ξ)θ(γ)}{θ(u − γ)θ(ξ)} .

Let R(u, λ) be the elliptic dynamical R-matrix [F]:

(1.3)  R(u, λ) = ∑_{a=1}^{N} E_{aa} ⊗ E_{aa} + ∑_{{a,b=1 \atop a≠b}}^{N}(α(u, λ_{ab})E_{aa} ⊗ E_{bb} + β(u, λ_{ab})E_{ab} ⊗ E_{ba}) ,

where λ ∈ 𝔥^{∗} and λ_{ab} = λ_{a} − λ_{b}. The dynamical R-matrix has zero weight:

(1.4)  [R(u, λ) , h^{(1)} + h^{(2)}] = 0 ,

satisfies the inversion relation:

(1.5)  R(u, λ)R^{(21)}(−u, λ) = 1 ,

and the dynamical Yang-Baxter equation:

(1.6)  R^{(12)}(u − v, λ − γh^{(3)})R^{(13)}(u, λ)R^{(23)}(v, λ − γh^{(1)}) =
= R^{(23)}(v, λ)R^{(13)}(u, λ − γh^{(2)})R^{(12)}(u − v, λ) .

The last equality holds in End(ℂ^{N} ⊗ ℂ^{N} ⊗ ℂ^{N}) . By standard convention, we assume that R^{(ij)}(u, λ) acts as R(u, λ) in the i-th and j-th tensor factors and as the identity operator in the remaining factors. For instance, in formula (1.6) we have R^{(12)} = R ⊗ id and R^{(23)} = id ⊗ R. Notice that for the R-matrix (1.3) we in addition have

R(u, λ − γ (h^{(1)}+ h^{(2)})) = R(u, λ) .

2. Elliptic quantum group E_{τ,γ}(𝔰𝔩_{N})

A module over the elliptic quantum group E_{τ,γ}(𝔰𝔩_{N}) is a diagonalizable 𝔥-module V together with D(V)-valued meromorphic functions T_{ab}(u) , a, b = 1, …, N , of a complex variable u, subject to relations (2.1) – (2.3). We combine the functions T_{ab}(u) into a matrix T (u) with noncommuting entries:

T (u) = ∑_{a,b} E_{ab} ⊗ T_{ab}(u) .

The defining relations are:

(2.1)  T_{ab}(u) φ(λ) = φ(λ − γ ε_{b}) T_{ab}(u)

for any φ ∈ Fun(ℂ) ,

[T (u) , h^{(1)} + h^{(2)}] = 0 , (2.2)

R^{(12)}(u − v, λ − γh^{(3)})T^{(13)}(u)T^{(23)}(v) = T^{(23)}(v)T^{(13)}(u)R^{(12)}(u − v, λ) . (2.3)

The last equality holds in End(ℂ^{N} ⊗ ℂ^{N} ⊗ Fun(V)). Here T^{(13)}(u) = ∑_{a,b} E_{ab} ⊗ id ⊗ T_{ab}(u) and

T^{(23)}(u) = ∑_{a,b} id ⊗ E_{ab} ⊗ T_{ab}(u) .

Relations (2.1) can be written as follows: T (u) φ(λ + γh^{(1)}) = φ(λ) T (u) for any φ ∈ Fun(ℂ) . Formula (2.2) means that for any μ ∈ 𝔥^{∗}

(2.4)  T_{ab}(u) Fun(V [μ]) ⊂ Fun(V [μ − ε_{a} + ε_{b}]) .

Introduce the quantum determinant Det T (u) , cf. [FV1], [FV2], by the rule

(2.5)  Det T (u) = \frac{Θ(λ)}{Θ(λ − γh)} ∑_{i∈S_{N}}sign(i) T_{i_{N},N}(u + (N − 1)γ) … T_{i_{2},2}(u + γ) T_{i_{1},1}(u)

where Θ(λ) = ∏_{1⩽a<b⩽N}θ(λ_{a} − λ_{b}) , the sum is taken over all permutations i = (i_{1}, …, i_{N}) and sign(i)

is the sign of the permutation. It is clear that Det T (u) commutes with multiplication by any function φ(λ) ∈ Fun(ℂ) and with the action of 𝔥. Hence, Det T (u) acts in Fun(V) as multiplication by some End(V)-valued meromorphic function of u and λ. We denote this function by Det L(u, λ) , cf. (2.6).

Proposition 2.1. [Det T (u), T_{ab}(v)] = 0 for any a, b = 1, …, N .

The proposition is proved in Appendix B.

According to (2.1), T_{ab}(u) is a difference operator; for any v ∈ Fun(V) we have

(2.6)  (T_{ab}(u) v)(λ) = L_{ab}(u, λ) v(λ − γ ε_{b})

where L_{ab}(u, λ) ∈ Fun(End(V)). Set L(u, λ) = ∑_{a,b} E_{ab} ⊗ L_{ab}(u, λ) .

Example. For any x ∈ ℂ the assignment L(u, λ) ↦ R(u − x, λ) makes ℂ^{N} into an E_{τ,γ}(𝔰𝔩_{N})-module. The module is called the vector representation of E_{τ,γ}(𝔰𝔩_{N}) with the evaluation point x. The quantum determinant in this E_{τ,γ}(𝔰𝔩_{N})-module is

Det L(u, λ) = \frac{θ(u − x + (N − 1)γ)}{θ(u − x)} .

Let V, W be E_{τ,γ}(𝔰𝔩_{N})-modules. An element φ ∈ Fun(Hom_{𝔥}(V, W)) is called a morphism of E_{τ,γ}(𝔰𝔩_{N})-modules if the induced map Fun(V) → Fun(W) satisfies

φ(λ) T_{ab}(u)|_{Fun(V)} = T_{ab}(u)|_{Fun(W)} φ(λ)

for any a, b = 1, …, N . Denote by Mor(V, W) the space of all morphisms from V to W. A morphism φ is called an isomorphism if the map φ(λ) is bijective for generic λ.

An E_{τ,γ}(𝔰𝔩_{N})-module V is called irreducible if for any nontrivial morphism φ ∈ Mor(W, V) the map map φ(λ) is surjective for generic λ, and reducible, otherwise.

Let V, W be E_{τ,γ}(𝔰𝔩_{N})-modules. Then the 𝔥-module V ⊗ W is made into an E_{τ,γ}(𝔰𝔩_{N})-module by the rule

(2.7)  L_{ab}(u, λ)|_{V ⊗W} = ∑_{c=1}^{N} L_{ac}(u, λ − γh^{(2)}) ⊗ L_{cb}(u, λ) ,

and T_{ab}(u) acts in Fun(V ⊗W) according to (2.6). Triple tensor products (U ⊗ V) ⊗ W and U ⊗ (V ⊗ W) are canonically isomorphic as E_{τ,γ}(𝔰𝔩_{N})-modules. The quantum determinant is group-like, it acts in the E_{τ,γ}(𝔰𝔩_{N})-module V ⊗ W by

Det L(u, λ)|_{V ⊗W} = Det L(u, λ − γh^{(2)}) ⊗ Det L(u, λ) .

If φ_{1}, φ_{2} are morphisms of E_{τ,γ}(𝔰𝔩_{N})-modules: φ_{1} ∈ Mor(V_{1}, W_{1}) , φ_{2} ∈ Mor(V_{2}, W_{2}) , then

φ_{1}(λ − γh^{(2)}) ⊗ φ_{2}(λ)

is a morphism of E_{τ,γ}(𝔰𝔩_{N})-modules V_{1} ⊗ V_{2} and W_{1} ⊗ W_{2} .

Remark. Notice that the given definition of modules over the elliptic quantum group E_{τ,γ}(𝔰𝔩_{N}) is slightly different from the corresponding definition in [FV1]. The definition of morphisms of E_{τ,γ}(𝔰𝔩_{N})-modules is also suitably modified. We choose the present version in order to simplify the exposition.

3. Small elliptic quantum group e_{τ,γ}(𝔰𝔩_{N})

Let Fun^{⊗2}(ℂ) be the ring of meromorphic functions f(λ^{{1}}, λ^{{2}}) on 𝔥^{∗}⊕ 𝔥^{∗} such that location of singularities of f(λ^{{1}}, λ^{{2}}) in λ^{{1}} does not depend on λ^{{2}} and vice versa. For brevity, we write f(λ^{{1}}) or f(λ^{{2}}) instead of f(λ^{{1}}, λ^{{2}}) if the function does not depend on the other variable.

Given a diagonalizable 𝔥-module V we define an action of Fun^{⊗2}(ℂ) in Fun(V) : for any f ∈ Fun^{⊗2}(ℂ) set f : v(λ) ↦ f(λ, λ − γh) v(λ) . For instance, for any φ ∈ Fun(ℂ) we have

(3.1)  φ(λ^{{1}}) : v(λ) ↦ φ(λ) v(λ) ,  φ(λ^{{2}}) : v(λ) ↦ φ(λ − γh) v(λ) .

We always assume that Fun^{⊗2}(ℂ) acts in Fun(V) in this way.

The operator algebra eτ,γ^{O}(𝔰𝔩_{N}) is a unital associative algebra over ℂ generated by elements t_{ab}, a, b = 1, …, N , and functions f ∈ Fun^{⊗2}(ℂ) subject to relations

(3.2)  t_{ab} f(λ^{{1}}, λ^{{2}}) = f(λ^{{1}}− γ ε_{a} , λ^{{2}}− γ ε_{b}) t_{ab}

for any f ∈ Fun^{⊗2}(ℂ) ,

t_{ab} t_{ac} = t_{ac} t_{ab} , (3.3)

t_{ac} t_{bc} = \frac{θ(λ_{ab}^{\{1\}}+ γ)}{θ(λ_{ab}^{\{1\}}− γ)} t_{bc} t_{ac} , for a ≠ b, (3.4)

\frac{θ(λ_{bd}^{\{2\}}+ γ)}{θ(λ_{bd}^{\{2\}})} t_{ab} t_{cd} − \frac{θ(λ_{ac}^{\{1\}}+ γ)}{θ(λ_{ac}^{\{1\}})} t_{cd} t_{ab} = \frac{θ(λ_{ac}^{\{1\}}+ λ_{bd}^{\{2\}}) θ(γ)}{θ(λ_{ac}^{\{1\}}) θ(λ_{bd}^{\{2\}})} t_{ad} t_{cb} , (3.5)

for a ≠ c and b ≠ d. Here λ_{ab}^{{i}} = λ_{a}^{{i}}− λ_{b}^{{i}}.

The ring Fun^{⊗2}(ℂ) is embedded into eτ,γ^{O}(𝔰𝔩_{N}) as a commutative subalgebra. It acts on eτ,γ^{O}(𝔰𝔩_{N}) by left multiplication. In this paper we consider eτ,γ^{O}(𝔰𝔩_{N}) as the corresponding Fun^{⊗2}(ℂ)-module.

The operator algebras eτ,γ^{O}(𝔰𝔩_{N}) , eτ+1,γ^{O}(𝔰𝔩_{N}) and e−1/τ,−γ/τ^{O} (𝔰𝔩_{N}) are isomorphic. The isomorphism eτ,γ^{O}(𝔰𝔩_{N}) → eτ+1,γ^{O}(𝔰𝔩_{N}) corresponds to the property θ(u; τ) = e^{πi/4} θ(u; τ + 1) of the theta function and is tautological. The isomorphism eτ,γ^{O}(𝔰𝔩_{N}) → e−1/τ,−γ/τ^{O} (𝔰𝔩_{N}) corresponds to the equality

θ(u; −1/τ) = (iτ)^{1/2} exp(πiτ u^{2}) θ(−τ u; τ) , Im(iτ)^{1/2} > 0 ,

and is given by the following formulae

f(λ^{{1}}, λ^{{2}}) ↦ f(−τ λ^{{1}}, −τ λ^{{2}}) ,

t_{ab} ↦ exp( \frac{πiτ}{2N} ((N + 1) ∑_{c=1}^{N}(λ_{ca}^{\{1\}})^{2} − (N − 1) ∑_{c=1}^{N}(λ_{cb}^{\{2\}})^{2} − 4N λ_{a}^{\{1\}}λ_{b}^{\{2\}} )) t_{ab} .

Introduce a matrix T (u) with noncommuting entries:

(3.6)  T (u) = ∑_{a,b} E_{ab} ⊗ T_{ab}(u) ,  T_{ab}(u) = θ(u − λ_{b}^{{1}}+ λ_{a}^{{2}}) t_{ba} .

Theorem 3.1. The commutation relations (3.3) – (3.5) are equivalent to

(3.7)  R^{(12)}(u − v, λ^{{2}})T^{(13)}(u)T^{(23)}(v) = T^{(23)}(v)T^{(13)}(u)R^{(12)}(u − v, λ^{{1}}) ,

The proof is straightforward and is based on summation formulae for the theta function. Notice that formula (3.7) is similar to (2.3).

Introduce the quantum determinant Det T (u) like in (2.5):

(3.8)  Det T (u) = \frac{Θ(λ^{\{1\}})}{Θ(λ^{\{2\}})} ∑_{i∈S_{N}}sign(i) T_{i_{N},N}(u + (N − 1)γ) … T_{i_{2},2}(u + γ) T_{i_{1},1}(u)

where Θ(λ) = ∏_{1⩽a<b⩽N}θ(λ_{a} − λ_{b}) . By (3.6) and (3.2) we obtain

(3.9)  Det T (u) = \frac{Θ(λ^{\{1\}})}{Θ(λ^{\{2\}})} ∑_{i∈S_{N}}sign(i) ∏_{a=1}^{N} θ(u + (a − 1)γ − λ_{a}^{\{1\}}+ λ_{i_{a}}^{\{2\}}) t_{N,i_{N}} … t_{1,i_{1}} .

It is clear that Det T (u) f(λ^{{1}}, λ^{{2}}) = f(λ^{{1}}, λ^{{2}}) Det T (u) for any f ∈ Fun^{⊗2}(ℂ) .

Proposition 3.2. [Det T (u), t_{ab}] = 0 for any a, b = 1, …, N .

The proof is similar to the proof of Proposition 2.1.

Let f̄ = (f_{1}, …, f_{N}) be a multiplicative cocycle with coefficients in Fun(ℂ) , that is, the functions f_{1}, …, f_{N} ∈ Fun(ℂ) satisfy the condition

f_{a}(λ) f_{b}(λ − γ ε_{a}) = f_{a}(λ − γ ε_{b}) f_{b}(λ)

for any a, b = 1, …, N , cf. Appendix C. Then the assignments

(3.10)  t_{ab} ↦ f_{a}(λ^{{1}}) t_{ab}  and  t_{ab} ↦ f_{b}(λ^{{2}}) t_{ab}

define two endomorphisms of the operator algebra eτ,γ^{O}(𝔰𝔩_{N}) . The endomorphisms are automorphisms if f̄ is nondegenerate, that is, if f_{a} ≠ 0 for any a = 1, …, N . The automorphisms are inner if f̄ is a multiplicative coboundary: f_{a}(λ) = φ(λ − γ ε_{a})/φ(λ) for a certain function φ ∈ Fun(ℂ) , see (3.2).

Remark. In this paper the inequality f ≠ 0 for a meromorphic function f means that the function f is not identically zero.

Proposition 3.3. The assignment

f(λ^{{1}}, λ^{{2}}) ↦ f(−λ^{{2}}, −λ^{{1}}) , (3.11)

t_{ab} ↦ ∏_{{c=1 \atop c≠b}}^{N} θ(λ_{cb}^{\{1\}}) ∏_{{c=1 \atop c≠a}}^{N} (θ(λ_{ca}^{\{2\}}+ γ))^{−1} t_{ba}

defines an involutive antiautomorphism of the operator algebra eτ,γ^{O}(𝔰𝔩_{N}) .

A module over the small elliptic quantum group e_{τ,γ}(𝔰𝔩_{N}) is a diagonalizable 𝔥-module V endowed with an action of the operator algebra eτ,γ^{O}(𝔰𝔩_{N}) in the space Fun(V) . Relations (3.1) and (3.2) mean that t_{ab} acts in Fun(V) as a difference operator:

(3.12)  (t_{ab} v)(λ) = ℓ_{ab}(λ)v(λ − γ ε_{a})  for any v ∈ Fun(V)

where ℓ_{ab}(λ) ∈ Fun(End(V)) is a suitable function. In D(V) relations (3.1), (3.2) are equivalent to

t_{ab} φ(λ) = φ(λ − γ ε_{a}) t_{ab}

for any φ ∈ Fun(ℂ) , and

t_{ab} Fun(V [μ]) ⊂ Fun(V [μ + ε_{a} − ε_{b}])

for any μ ∈ 𝔥^{∗}, which are similar to (2.1) and (2.4). For the quantum determinant we have that

(Det T (u) v)(λ) = D(u, λ)v(λ)

for any v ∈ Fun(V) where D(u, λ) ∈ Fun(End(V)) is a suitable function.

Example. The assignment

(3.13)  ℓ_{aa}(λ) ↦ E_{aa} + ∑_{{b=1 \atop b≠a}}^{N} \frac{θ(λ_{ab} + γ)}{θ(λ_{ab})} E_{bb} ,

ℓ_{ab}(λ) ↦ \frac{θ(γ)}{θ(λ_{ab})} E_{ab} ,  a ≠ b ,

a, b = 1, …, N , makes ℂ^{N} into an e_{τ,γ}(𝔰𝔩_{N})-module. The module is called the vector representation of e_{τ,γ}(𝔰𝔩_{N}) . In the vector representation

D(u, λ) = θ(u − γ) θ(u + γ) θ(u + 2γ) … θ(u + (N − 1)γ) .

Corollary 3.4. For any e_{τ,γ}(𝔰𝔩_{N})-module V and any x ∈ ℂ the rule T_{ab}(u) = T_{ab}(u − x)|_{V} makes V into an E_{τ,γ}(𝔰𝔩_{N})-module called the evaluation module V (x) over E_{τ,γ}(𝔰𝔩_{N}) .

By abuse of notation we call the assignment T (u) ↦ T (u) the evaluation morphism E_{τ,γ}(𝔰𝔩_{N}) → e_{τ,γ}(𝔰𝔩_{N}) . It is analogous to the evaluation homomorphism from the Yangian Y (𝔰𝔩_{N}) to U(𝔰𝔩_{N}) .

Remark. Corollary 3.4 was our main motivation to discover and study the small elliptic quantum group e_{τ,γ}(𝔰𝔩_{N}) .

Let V, W be e_{τ,γ}(𝔰𝔩_{N})-modules. An element φ ∈ Fun(Hom_{𝔥}(V, W)) is a morphism of e_{τ,γ}(𝔰𝔩_{N}) modules if the induced map intertwines the corresponding actions of eτ,γ^{O}(𝔰𝔩_{N}) :

φ(λ) t_{ab}|_{Fun(V)} = t_{ab}|_{Fun(W)} φ(λ)

for any a, b = 1, …, N . Denote by Mor(V, W) the space of all morphisms from V to W. A morphism φ is called an isomorphism if the map φ(λ) is bijective for generic λ.

An e_{τ,γ}(𝔰𝔩_{N})-module V is called irreducible if for any nontrivial morphism φ ∈ Mor(W, V) the map φ(λ) is surjective for generic λ, and reducible, otherwise.

Say that an e_{τ,γ}(𝔰𝔩_{N})-module W is a submodule of V if there is a morphism φ ∈ Mor(W, V) such that the map φ(λ) is injective for generic λ. The morphism φ is called an embedding. The submodule W is called proper if φ is not an isomorphism. Any e_{τ,γ}(𝔰𝔩_{N})-module V has at least two submodules: V itself and the trivial submodule {0} with obvious embeddings.

Let W be a submodule of V . Then one can define the quotient e_{τ,γ}(𝔰𝔩_{N})-module V/W as follows. Fix an embedding φ. The subspace φ(Fun(W)) ⊂ Fun(V) is invariant with respect to the action of eτ,γ^{O}(𝔰𝔩_{N}) , hence eτ,γ^{O}(𝔰𝔩_{N}) acts on Fun(V)/φ(Fun(W)). Let λ_{0} ∈ 𝔥^{∗} be such that the map φ(λ_{0}) is injective. Take a complement U of φ(λ_{0})W in V , that is, V = U ⊕ φ(λ_{0})W as a vector space. Notice that V = U ⊕ φ(λ)W for generic λ as well. Then Fun(V) = Fun(U) ⊕ φ(Fun(W)) and, therefore, Fun(U) = Fun(V)/φ(Fun(W)), which induces an action of eτ,γ^{O}(𝔰𝔩_{N}) on Fun(U) and makes U into an e_{τ,γ}(𝔰𝔩_{N})-module. The constructed e_{τ,γ}(𝔰𝔩_{N})-module does not depend on a choice φ, λ_{0} and U up to an isomorphism of e_{τ,γ}(𝔰𝔩_{N})-modules and is denoted by V/W.

Lemma 3.5. An e_{τ,γ}(𝔰𝔩_{N})-module V is reducible if and only if it has a nontrivial proper submodule.

Lemma 3.6. An e_{τ,γ}(𝔰𝔩_{N})-module V is irreducible if and only if for any nontrivial morphism φ ∈ Mor(V, W) the map φ(λ) is injective for generic λ.

4. Highest weight modules over e_{τ,γ}(𝔰𝔩_{N})

For any monomial t_{a_{1}b_{1}} … t_{a_{k}b_{k}} set deg(t_{a_{1}b_{1}} … t_{a_{k}b_{k}}) = k. For k = 0 we assume that the monomial equals 1 . The operator algebra eτ,γ^{O}(𝔰𝔩_{N}) is generated by all monomials t_{a_{1}b_{1}} … t_{a_{k}b_{k}} , k = 0, 1, …, as a Fun^{⊗2}(ℂ)-module.

For any function φ ∈ Fun^{⊗2}(ℂ) set deg(φ) = 0 . Since relations (3.2) – (3.5) are homogeneous, the algebra eτ,γ^{O}(𝔰𝔩_{N}) is ℤ_{⩾0} -graded by deg . Let 𝔢_{k} = {x ∈ eτ,γ^{O}(𝔰𝔩_{N}) | deg(x) = k} be the homogeneous subspace of degree k. Each subspace 𝔢_{k} is finitely generated over Fun^{⊗2}(ℂ) .

Consider the normal ordering of generators: t_{ab} < t_{cd} if a − b < c − d, or a − b = c − d and a < c. Say that the monomial t_{a_{1}b_{1}} … t_{a_{k}b_{k}} is normally ordered if t_{a_{i}b_{i}} < t_{a_{j}b_{j}} for any i < j, or k = 0 .

Theorem 4.1. For any k ∈ ℤ_{⩾0} the normally ordered monomials of degree k form a basis of 𝔢_{k} over Fun^{⊗2}(ℂ) .

Proof. For k = 0 and k = 1 the claim is immediate. Let k > 1 . Here we prove that the normally ordered monomials of degree k span 𝔢_{k} over Fun^{⊗2}(ℂ) . The linear independence of the normally ordered monomials is proved in Appendix D.

It is clear from relations (3.3) – (3.5) that any product t_{ab} t_{cd} can be written as a linear combination of normally ordered products. Given a monomial t_{a_{1}b_{1}} … t_{a_{k}b_{k}} we take any disordered product of adjacent factors and replace it by a suitable sum of normally ordered products, then do the same for each of the obtained monomials. To see that the procedure terminates and, hence, produces a linear combination of normally ordered monomials, introduce auxilary gradings on monomials by the rule

r(t_{a_{1}b_{1}} … t_{a_{k}b_{k}}) = ∑_{i=1}^{k} i(a_{i} − b_{i}) , r′(t_{a_{1}b_{1}} … t_{a_{k}b_{k}}) = ∑_{i=1}^{k} ib_{i} ,

and observe that at each nontrivial step of the procedure we replace a monomial by a sum of monomials of either less degree r, or the same degree r and less degree r′. □

Introduce modified generators of the algebra eτ,γ^{O}(𝔰𝔩_{N}) . For any a, b = 1, …, N set

(4.1)  t̂_{ab} = ∏_{1⩽c<a}θ(λ_{ca}^{{1}}) ∏_{1⩽c<b}(θ(λ_{cb}^{{2}}))^{−1} t_{ab} .

Let V be an e_{τ,γ}(𝔰𝔩_{N})-module. A nonzero function v ∈ Fun(V) is called a singular vector if t_{ab} v = 0 for any 1 ⩽ a < b ⩽ N . Say that v is a regular singular vector if in addition v is a weight vector with respect to the action of 𝔥 and

(4.2)  (t̂_{aa}v)(λ) = Q_{a}(λ) v(λ) ,  a = 1, …, N ,

for certain functions Q_{1}, …, Q_{N} ∈ Fun(ℂ) . We call Q̂ = (Q_{1}, …, Q_{N}) the dynamical weight of v. Relation (A.2) implies that Q̂ is a multiplicative cocycle:

Q_{a}(λ) Q_{b}(λ − γ ε_{a}) = Q_{a}(λ − γ ε_{b}) Q_{b}(λ)

for any a, b = 1, …, N . If f ∈ Fun(ℂ) , then the function ṽ(λ) = f(λ) v(λ) is a regular singular vector of dynamical weight (Q_{1}, …, Q_{N}) where

Q_{a}(λ) = Q_{a}(λ) \frac{f(λ − γ ε_{a})}{f(λ)} , a = 1, …, N .

Hence, the subspace Fun(ℂ) v determines the dynamical weight up to a multiplicative coboundary.

Say that Q̂ and v are nondegenerate if Q_{a} ≠ 0 for any a = 1, …, N . Say that Q̂ and v are standard if Q_{a} = 1 for any a = 1, …, N .

By formula (3.9) the quantum determinant acts on a regular singular vector v of weight μ and dynamical weight Q̂ as follows:

(4.3)  (Det T (u) v)(λ) = ∏_{a=1}^{N} θ(u − γ (μ_{a} − a + 1)) ∏_{a=1}^{N} Q_{a}(λ − ∑_{a<b⩽N}γ ε_{b}) v(λ) .

An e_{τ,γ}(𝔰𝔩_{N})-module V is called a highest weight module with highest weight μ, dynamical highest weight Q̂ and highest weight vector v if v is a regular singular vector of weight μ and dynamical weight Q̂ generating Fun(V) over eτ,γ^{O}(𝔰𝔩_{N}) . If Q̂ is standard (nondegenerate), then V is called a standard (nondegenerate) e_{τ,γ}(𝔰𝔩_{N})-module of highest weight μ. For example, the vector representation is a standard e_{τ,γ}(𝔰𝔩_{N})-module of highest weight ω_{1} .

It is clear that any nondegenerate highest weight module is isomorphic to a pullback of a standard highest weight module of the same highest weight through a suitable automorphism of the form (3.10).

Proposition 4.2. Let V be a highest weight module with highest weight μ. Then

a) V = ⨁_{ν⩽μ}V [ν] and dim_{ℂ} V [μ] = 1 ;

b) V is reducible if and only if it has a singular vector of weight ν < μ.

Lemma 4.3. Let V be a highest weight module of highest weight μ and dynamical highest weight (Q_{1}, …, Q_{N}) . Then for any v ∈ Fun(V)

(4.4)  (Det T (u) v)(λ) = ∏_{a=1}^{N} θ(u − γ (μ_{a} − a + 1)) ∏_{a=1}^{N} Q_{a}(λ − ∑_{a<b⩽N}γ ε_{b}) v(λ) .

The statement follows from Proposition 3.2 and formula (4.3).

Corollary 4.4. Let γ ∉ ℚ + τ ℚ. Then a nondegenerate highest weight module V of highest weight μ is irreducible unless V [w ·μ] ≠ 0 for some w ∈ W such that w ·μ < μ. Here W is the Weil group.

Proof. If V is reducible, then it has a singular vector v of weight ν < μ. Comparing formulae (4.3) and (4.4) for the action of Det T (u) on v we obtain that (ν_{1} − 1, …, ν_{N} − N) = (μ_{i_{1}}− i_{1}, …, μ_{i_{N}} − i_{N}) for some permutation (i_{1}, …, i_{N}) since γ ∉ ℚ + τ ℚ. That is, ν = w(μ + ρ) − ρ = w ·μ for some w ∈ W because ρ = − ε_{1} − 2ε_{2} − … − Nε_{N} . □

Let e(𝔟_{+}) and e(𝔫_{+}) be the left ideals in eτ,γ^{O}(𝔰𝔩_{N}) generated by the elements t_{ab} with a ⩽ b and a < b, respectively. Let B , B′, N , N′ be the following sets of normally ordered monomials

(4.5)  B = {t̂_{a_{1}b_{1}} … t̂_{a_{k}b_{k}} | k ⩾ 0 , a_{i} ⩾ b_{i} , i = 1, …, k} ,
B′ = {t̂_{a_{1}b_{1}} … t̂_{a_{k}b_{k}} | k > 0 , a_{i} ⩽ b_{i} , i = 1, …, k} ,
N = {t̂_{a_{1}b_{1}} … t̂_{a_{k}b_{k}} | k ⩾ 0 , a_{i} > b_{i} , i = 1, …, k} ,
N′ = {t̂_{a_{1}b_{1}} … t̂_{a_{k}b_{k}} | k > 0 , a_{i} < b_{i} , i = 1, …, k} .

Lemma 4.5. The normally ordered monomials of the form mm′ where m ∈ N and m′ ∈ B′ form a basis of e(𝔟_{+}) over Fun^{⊗2}(ℂ) . Lemma 4.6. The normally ordered monomials of the form mm′ where m ∈ B and m′ ∈ N′ form a basis of e(𝔫_{+}) over Fun^{⊗2}(ℂ) . The statements easily follows from relations (3.3) – (3.5) and Theorem 4.1.

For any monomial t_{a_{1}b_{1}} … t_{a_{k}b_{k}} set wt(t_{a_{1}b_{1}} … t_{a_{k}b_{k}}) = ∑_{i=1}^{k}(ε_{a_{i}} − ε_{b_{i}}) and for any function φ ∈ Fun^{⊗2}(ℂ) set wt(φ) = 0 . Since relations (3.2) – (3.5) are homogeneous, the algebra eτ,γ^{O}(𝔰𝔩_{N}) is ℙ graded by wt .

Let μ ∈ 𝔥^{∗} and Q̂ be a multiplicative cocycle. Below we define a Verma module M_{μ,Q̂} of highest weight μ and dynamical highest weight Q̂ over e_{τ,γ}(𝔰𝔩_{N}) .

Let N_{ℂ} = ⨁_{m ∈ N} ℂm be a diagonalizable 𝔥-module such that a monomial m has weight wt(m) , and let ℂv_{μ,Q̂} be an 𝔥-module such that v_{μ,Q̂} has weight μ. Then M_{μ,Q̂} = N_{ℂ} ⊗ ℂv_{μ,Q̂} as an 𝔥-module. We define an action of eτ,γ^{O}(𝔰𝔩_{N}) in Fun(M_{μ,Q̂}) by the rule: 1 ⊗ v_{μ,Q̂} is a regular singular vector of weight μ and dynamical weight Q̂ and

m (1 ⊗ v_{μ,Q̂}) = m ⊗ v_{μ,Q̂}

for any m ∈ N . This determines an action on v_{μ,Q̂} by any normally ordered monomial and, hence, by any element of eτ,γ^{O}(𝔰𝔩_{N}) , cf. Theorem 4.1. Finally, for any x ∈ eτ,γ^{O}(𝔰𝔩_{N}) set

x (m ⊗ v_{μ,Q̂}) = (xm) (1 ⊗ v_{μ,Q̂})

where the product xm should be represented as a linear combination of normally ordered monomials. Proposition 4.7. M_{μ,Q̂} is a well-defined e_{τ,γ}(𝔰𝔩_{N})-module with highest weight μ, dynamical highest weight Q̂ and highest weight vector 1 ⊗ v_{μ,Q̂}. The statement follows from Lemmas 4.5, 4.6 and Theorem 4.1.

From now on we suppress the symbol of tensor product in the definition of the Verma module M_{μ,Q̂}, for instance, we write v_{μ,Q̂} instead of 1 ⊗ v_{μ,Q̂}.

If Q̂ is standard, then the e_{τ,γ}(𝔰𝔩_{N})-module M_{μ} = M_{μ,Q̂} is called the standard Verma module with highest weight μ and the highest weight vector v_{μ} = v_{μ,Q̂}. Lemma 4.8. Let V be an e_{τ,γ}(𝔰𝔩_{N})-module. For any v ∈ Fun(V) which is a regular singular vector of weight μ and dynamical weight Q̂, there is a unique morphism φ ∈ Mor(M_{μ,Q̂} , V) which sends v_{μ,Q̂} to v. Proposition 4.9. Any highest weight e_{τ,γ}(𝔰𝔩_{N})-module is isomorphic to a suitable quotient of the Verma module over e_{τ,γ}(𝔰𝔩_{N}) of the same highest weight and dynamical highest weight. Proposition 4.10. For any μ ∈ 𝔥^{∗} and a dynamical weight Q̂ there exists a unique up to an isomorphism irreducible highest weight e_{τ,γ}(𝔰𝔩_{N})-module with highest weight μ and dynamical highest weight Q̂. Proposition 4.11. Let γ ∉ ℚ + τ ℚ. Then a highest weight e_{τ,γ}(𝔰𝔩_{N})-module with highest weight μ and a nondegenerate dynamical highest weight Q̂ is isomorphic to the Verma module M_{μ,Q̂} unless w ·μ < μ for some w ∈ W.

5. Dynamical Shapovalov form

Let e(𝔫_{−}) be the right ideal in eτ,γ^{O}(𝔰𝔩_{N}) generated by the elements t_{ab} with a > b, let 𝔡 be the Fun^{⊗2}(ℂ)-submodule generated by normally ordered monomials of the form t_{a_{1}a_{1}} … t_{a_{k}a_{k}} , and let

𝔢[0] = {x ∈ eτ,γ^{O}(𝔰𝔩_{N}) | wt(x) = 0}

be the subalgebra of zero weight elements in eτ,γ^{O}(𝔰𝔩_{N}) . Consider the quotient

e(𝔥) = eτ,γ^{O}(𝔰𝔩_{N})/(e(𝔫_{+}) + e(𝔫_{−})) .

Let η : eτ,γ^{O}(𝔰𝔩_{N}) → e(𝔥) be the natural projection. By Theorem 4.1 the restriction of η to 𝔡 is a bijection. Denote by η̄ : e(𝔥) → 𝔡 the inverse map. For any x, y ∈ e(𝔥) define their product by the rule xy = η(η̄(x)η̄(y)). It is easy to see that this defines an algebra structure on e(𝔥) . Lemma 5.1. The restriction of η to 𝔢[0] is a homomorphism.

Set q_{a} = η(t̂_{aa}) , a = 1, …, N . It follows from (A.2) that e(𝔥) is generated by functions f ∈ Fun^{⊗2}(ℂ) and the pairwise commuting elements q_{1}, …, q_{N} subject to relations

(5.1)  q_{a} f(λ^{{1}}, λ^{{2}}) = f(λ^{{1}}− γ ε_{a} , λ^{{2}}− γ ε_{a}) q_{a} .

The assignment ϖ : f(λ^{{1}}, λ^{{2}}) ↦ f(−λ^{{2}}, −λ^{{1}}) ,

ϖ : t_{ab} ↦ ∏_{1⩽c<b}θ(λ_{cb}^{{1}}) θ(λ_{bc}^{{1}}− γ) ∏_{1⩽c<a}(θ(λ_{ca}^{{2}}) θ(λ_{ac}^{{2}}− γ))^{−1} t_{ba}

defines an involutive antiautomorphism of the operator algebra eτ,γ^{O}(𝔰𝔩_{N}) , which differs from the antiautomorphism (3.11) by a suitable automorphism of the form (3.10). We have

(5.2)  ϖ(t̂_{ab}) = t̂_{ba} .

For any x ∈ e(𝔥) set ϖ(x) = η(ϖ(η̄(x))). Lemma 5.2. For any m ∈ eτ,γ^{O}(𝔰𝔩_{N}) we have η(ϖ(m)) = ϖ(η(m)).

For any m_{1}, m_{2} ∈ eτ,γ^{O}(𝔰𝔩_{N}) set

(5.3)  S(m_{1} , m_{2}) = η(ϖ(m_{1})m_{2}) ∈ e(𝔥) .

Since ϖ is involutive, S(m_{1} , m_{2}) = ϖ(S(m_{2} , m_{1})). Moreover, S(m_{1}m_{2} , m_{3}) = S(m_{2} , ϖ(m_{1})m_{3}) for any m_{1}, m_{2}, m_{3} ∈ eτ,γ^{O}(𝔰𝔩_{N}) . S is called the dynamical Shapovalov form on eτ,γ^{O}(𝔰𝔩_{N}) .

Example. Let a < b. Then S(t̂_{ba} , t̂_{ba}) = − \frac{θ(λ_{ab}^{\{1\}}− λ_{ab}^{\{2\}}) θ(γ)}{θ(λ_{ab}^{\{1\}}) θ(λ_{ab}^{\{2\}})} q_{a} q_{b}.

Let μ ∈ 𝔥^{∗} and let Q̂ = (Q_{1}, …, Q_{N}) be a multiplicative cocycle. Then there is an algebra homomorphism χ_{μ,Q̂} : e(𝔥) → D(ℂ) :

χ_{μ,Q̂}(f(λ^{{1}}, λ^{{2}})) : φ(λ) ↦ f(λ, λ − γ μ) φ(λ) ,

χ_{μ,Q̂}(q_{a}) : φ(λ) ↦ Q_{a}(λ) φ(λ − γ ε_{a}) .

Consider the Verma module M_{μ,Q̂}. For any m ∈ eτ,γ^{O}(𝔰𝔩_{N}) and v ∈ Fun(M_{μ,Q̂}) set

(5.4)  S_{μ,Q̂}(m, v) = χ_{μ,Q̂}(S(m, m′))·1 ∈ Fun(ℂ) .

Here m′ ∈ eτ,γ^{O}(𝔰𝔩_{N}) is determined by v = m′v_{μ,Q̂} and 1 ∈ Fun(ℂ) is the constant function. It is easy to see that S_{μ,Q̂}(m, v) does not depend on a choice of m′. We call S_{μ,Q̂} the dynamical Shapovalov pairing for M_{μ,Q̂}. Lemma 5.3. For any m_{1}, m_{2} ∈ eτ,γ^{O}(𝔰𝔩_{N}) and v ∈ Fun(M_{μ,Q̂}) we have

S_{μ,Q̂}(m_{1} , m_{2} v) = S_{μ,Q̂}(ϖ(m_{2})m_{1} , v) = S_{μ,Q̂}(1, ϖ(m_{1})m_{2} v) .

Here 1 ∈ eτ,γ^{O}(𝔰𝔩_{N}) is the identity element.

Lemma 5.4. Let m ∈ eτ,γ^{O}(𝔰𝔩_{N}) be a wt-homogeneous element, and v ∈ M_{μ,Q̂}[ν] . If wt(m) ≠ μ − ν, then S_{μ,Q̂}(m, v) = 0 . Otherwise, ϖ(m) v = S_{μ,Q̂}(m, v) v_{μ,Q̂}. Example. S_{μ,Q̂}(1, v_{μ,Q̂}) = 1 . If a < b, then

S_{μ,Q̂}(t̂_{ba} , t̂_{ba}v_{μ,Q̂}) = − \frac{θ(γ μ_{ab}) θ(γ)}{θ(λ_{ab}) θ(λ_{ab} − γ μ_{ab})} Q_{a}(λ)Q_{b}(λ − γ ε_{a}) .

Set Ker S_{μ,Q̂} = {v ∈ Fun(M_{μ,Q̂}) | S_{μ,Q̂}(m, v) = 0 for any m ∈ eτ,γ^{O}(𝔰𝔩_{N})} .

The subspace Ker S_{μ,Q̂} is invariant under the action of eτ,γ^{O}(𝔰𝔩_{N}) and defines a proper submodule N_{μ,Q̂} of M_{μ,Q̂}. Proposition 5.5. N_{μ,Q̂} is the maximal proper submodule of M_{μ,Q̂}, that is, for any proper submodule U of M_{μ,Q̂} the embedded image of Fun(U) is contained in Fun(N_{μ,Q̂}) = Ker S_{μ,Q̂}. Proof. Let φ ∈ Mor(U, M_{μ,Q̂}) be the embedding. φ(Fun(U)) is a direct sum of its weight components and φ(Fun(U))[μ] = 0 since U is a proper submodule. Therefore, if v ∈ φ(Fun(U))[ν] and wt(m) = μ − ν, then ϖ(m)v = 0 . Hence, by Lemma 5.4 S_{μ,Q̂}(m, v) = 0 for any m ∈ eτ,γ^{O}(𝔰𝔩_{N}) and v ∈ φ(Fun(U)). □ Corollary 5.6. The quotient module V_{μ,Q̂} = M_{μ,Q̂}/N_{μ,Q̂} is the irreducible highest weight e_{τ,γ}(𝔰𝔩_{N}) module with highest weight μ and dynamical highest weight Q̂.

For any highest weight module V with highest weight μ, dynamical highest weight Q̂ and highest weight vector v one can define the Shapovalov pairing similarly to (5.4): for any m ∈ eτ,γ^{O}(𝔰𝔩_{N}) and v′ ∈ Fun(V) set

V S_{μ,Q̂}(m, v′) = χ_{μ,Q̂}(S(m, m′)) · 1 ∈ Fun(ℂ)

where m′ ∈ eτ,γ^{O}(𝔰𝔩_{N}) is determined by v′ = m′v. Propositions 4.9 and 5.5 imply that ^{V} S_{μ,Q̂}(m, v) does not depend on a choice of m′. Proposition 5.7. The module V is irreducible if and only if Ker ^{V} S_{μ,Q̂} is trivial. Otherwise, Ker ^{V} S_{μ,Q̂} defines the maximal proper submodule of V .

Let V_{μ} V ^{𝔰𝔩} be the irreducible 𝔰𝔩_{N}-module of highest weight μ. Set d_{μ}[ν] = dim_{ℂ} V_{μ} V ^{𝔰𝔩}[ν] . Theorem 5.8. Let μ ∈ 𝔥^{∗} be a dominant integral weight. Then dim_{ℂ} V_{μ,Q̂}[ν] ⩽ d_{μ}[ν] for any ν ∈ 𝔥^{∗}. In particular, the e_{τ,γ}(𝔰𝔩_{N})-module V_{μ,Q̂} is finite-dimensional. Theorem 5.9. Let γ ∉ ℚ + τ ℚ. The e_{τ,γ}(𝔰𝔩_{N})-module V_{μ,Q̂} for a nondegenerate dynamical weight Q̂ is finite-dimensional if and only if μ is a dominant integral weight. Moreover, dim_{ℂ} V_{μ,Q̂}[ν] = d_{μ}[ν] for any ν ∈ 𝔥^{∗}. Theorems 5.8 and 5.9 are proved in Section 8.

If Q̂ is standard, we set χ_{μ} = χ_{μ,Q̂}, S_{μ} = S_{μ,Q̂}, N_{μ} = N_{μ,Q̂} and V_{μ} = V_{μ,Q̂}.

6. Contragradient modules over e_{τ,γ}(𝔰𝔩_{N}) and contravariant form

For any diagonalizable 𝔥-module V define an involutive linear map ψ : Fun(V) → Fun(V) by the rule: if f ∈ Fun(V [μ]), then (ψf)(λ) = f(−λ + γμ) .

Let V be a diagonalizable 𝔥-module, and let V^{∗} = ⨁_{μ ∈ 𝔥^{∗}}(V [μ])^{∗} be its restricted dual space. We consider V^{∗} as a diagonalizable 𝔥-module such that (V [μ])^{∗} is a weight subspace of weight μ. For any B ∈ End(V) we denote by B^{∗} ∈ End(V^{∗}) the dual map. For a difference operator A ∈ D(V) we define the dual operator A′ ∈ D(V^{∗}) by the rule:

if (Av)(λ) = B(λ)v(λ + μ) , then (A′φ)(λ) = (B(λ − μ))^{∗}φ(λ − μ) ,

and the operator A^{†} ∈ D(V^{∗}) : A^{†} = ψ A′ψ. The assignment A ↦ A^{†} is an antihomomorphism D(V) → D(V^{∗}) .

Example. Let V be an e_{τ,γ}(𝔰𝔩_{N})-module. Then

((t_{ab})^{†}φ)(λ) = (ℓ_{ab}(−λ + (μ + ε_{b})γ))^{∗}φ(λ − γ ε_{b})

for any φ ∈ Fun(V^{∗}[μ]), cf. (3.12).

Given an e_{τ,γ}(𝔰𝔩_{N})-module V we make the 𝔥-module V^{∗} into an e_{τ,γ}(𝔰𝔩_{N})-module as follows: the action of an element m ∈ eτ,γ^{O}(𝔰𝔩_{N}) in Fun(V^{∗}) is given by (ϖ(m))^{†} where ϖ(m) is understood as a difference operator acting in Fun(V) . It is easy to check that the definition is consistent, that is, the ring Fun^{⊗2}(ℂ) acts in Fun(V^{∗}) in the prescribed way, cf. (3.1). The obtained e_{τ,γ}(𝔰𝔩_{N})-module V^{∗} is called the contragradient module to the e_{τ,γ}(𝔰𝔩_{N})-module V .

Consider the Verma module M_{μ,Q̂}. Recall that M_{μ,Q̂}[μ] = ℂv_{μ,Q̂}. Fix ^{∗} v_{μ,Q̂} ∈ ^{∗} M_{μ,Q̂}[μ] by the rule ⟨ ^{∗} v_{μ,Q̂} , v_{μ,Q̂}⟩ = 1 . For any a = 1, …, N set

Q_{a}(λ) = Q_{a}(−λ + (μ + ε_{a})γ) .

Notice that Q_{a}(λ) = Q_{a}(−λ + (μ + ε_{a})γ) as well.

Proposition 6.1. For the contragradient Verma module ^{∗} M_{μ,Q̂} the constant function ^{∗} v_{μ,Q̂} is a regular singular vector of weight μ and dynamical weight Q = (Q_{1}, …, Q_{N}) .

Corollary 6.2. There is a unique morphism π_{μ,Q̂} ∈ Mor(M_{μ,Q̂} , ^{∗} M_{μ,Q}) which sends v_{μ,Q̂} to ^{∗} v_{μ,Q}.

Theorem 6.3. Ker π_{μ,Q̂} = Ker S_{μ,Q̂}.

The morphism π_{μ,Q̂} induces a Fun(ℂ)-bilinear map B_{μ,Q̂} : Fun(M_{μ,Q̂}) ⊗_{ℂ} Fun(M_{μ,Q}) → Fun(ℂ) :

B_{μ,Q̂}(v , ṽ) = ⟨π_{μ,Q̂} v , ṽ⟩ .

Define a bilinear map C_{μ,Q̂} : Fun(M_{μ,Q̂}) ⊗_{ℂ} Fun(M_{μ,Q}) → Fun(ℂ) by the rule

(6.1)  C_{μ,Q̂}(v , ṽ) = B_{μ,Q̂}(v , ψṽ) .

The map C_{μ,Q̂} is called the contravariant form.

Theorem 6.4. Let v ∈ Fun(M_{μ,Q̂}[ν]) and ṽ ∈ Fun(M_{μ,Q}[ν̃]). Then C_{μ,Q̂}(v , ṽ) = 0 unless ν = ν̃. Moreover,

C_{μ,Q̂}(v , ṽ)(λ) = C_{μ,Q}(ṽ, v)(−λ + γν) .

Theorem 6.5.

a) C_{μ,Q̂}(v , ṽ) = 0 for any ṽ ∈ Fun(M_{μ,Q}) if and only if v ∈ Ker S_{μ,Q̂}.

b) C_{μ,Q̂}(v , ṽ) = 0 for any v ∈ Fun(M_{μ,Q̂}) if and only if ṽ ∈ Ker S_{μ,Q}.

Theorems 6.3 – 6.5 are proved at the end of the section.

Let V and Ṽ be highest weight e_{τ,γ}(𝔰𝔩_{N})-modules of the same highest weight μ and dynamical highest weights Q̂ and Q, respectively. By the last corollary the form C_{μ,Q̂} descends to a form Fun(V) ⊗_{ℂ} Fun(Ṽ) → Fun(ℂ) , which is denoted by the same letter. After obvious modification Theorem 6.5 remains true in this case. In particular, for the irreducible highest weight e_{τ,γ}(𝔰𝔩_{N})-modules V_{μ,Q̂} and V_{μ,Q} the corresponding function-valued bilinear form Fun(V_{μ,Q̂}) ⊗_{ℂ} Fun(V_{μ,Q}) → Fun(ℂ) is nondegenerate.

Example. C_{μ,Q̂}(v_{μ,Q̂} , v_{μ,Q}) = 1 . If a < b, then

C_{μ,Q̂}(t̂_{ba}v_{μ,Q̂} , t̂_{ba}v_{μ,Q}) = − \frac{θ(γ μ_{ab}) θ(γ)}{θ(λ_{ab} + γ) θ(λ_{ab} − γ μ_{ab} + γ)} Q_{a}(λ + γ ε_{a})Q_{b}(λ) .

For a monomial m = t̂_{a_{1}b_{1}} … t̂_{a_{k}b_{k}} set ζ′(m) = ∑_{i=1}^{k} ε_{a_{i}} and ζ′′(m) = ∑_{i=1}^{k} ε_{b_{i}} . Notice that wt(m) = ζ′(m) − ζ′′(m) .

Lemma 6.6. For any m_{1}, m_{2} ∈ eτ,γ^{O}(𝔰𝔩_{N}) we have

S_{μ,Q̂}(m_{1} , m_{2} v_{μ,Q̂})(λ) = S_{μ,Q}(m_{2} , m_{1}v_{μ,Q})(−λ + γ μ + γ ζ′(m_{1}) + ζ′′(m_{2})) .

Proof. Without loss of generality we can assume that ζ′(m_{1}) + ζ′′(m_{2}) = ζ′′(m_{1}) + ζ′(m_{2}) because expressions in both sides of the formula otherwise equal zero by Lemma 5.4.

Let m_{1} = t̂_{a_{1}b_{1}} … t̂_{a_{k}b_{k}} and m_{2} = t̂_{c_{1}d_{1}} … t̂_{c_{l}d_{l}} . Set (s_{1}, …, s_{k+l}) = (a_{1}, …, a_{k}, b_{1}, …, b_{l}) . By the definition of the Shapovalov pairing, cf. (5.4), for any multiplicative cocycle Q̂ we have

(6.2)  S_{μ,Q̂}(m_{1} , m_{2} v_{μ,Q̂})(λ) = S_{μ}(m_{1} , m_{2} v_{μ})(λ) ∏_{i=1}^{k+l} Q_{s_{i}}(λ − ∑_{1⩽j<i}γ ε_{s_{j}}) .

Here S_{μ} and v_{μ} correspond to the so-called standard case, cf. Section 5. Since Q̂ is a multiplicative cocycle, the product can be written also as ∏_{i} Q_{s_{i}}(λ − ∑_{j>i} γ ε_{s_{j}}). By formula (6.2) and the last remark it suffices to verify that

S_{μ}(m_{1} , m_{2} v_{μ})(λ) = S_{μ}(m_{2} , m_{1}v_{μ,Q})(−λ + γ μ + γ ζ′(m_{1}) + ζ′′(m_{2}))

which follows from the property S(m_{1} , m_{2}) = ϖ(S(m_{2} , m_{1})), commutation relations (5.1) and formula (5.4). □

Proposition 6.7. For any m_{1}, m_{2} ∈ N we have

(6.3)  C_{μ,Q̂}(m_{1}v_{μ,Q̂} , m_{2} v_{μ,Q})(λ) = S_{μ,Q̂}(m_{2} , m_{1}v_{μ,Q̂})(λ + γ ζ′′(m_{2})) .

Proof. Recall that by the definition of Verma modules m_{1}v_{μ,Q̂} and m_{2} v_{μ,Q} are constant functions since m_{1}, m_{2} ∈ N . If wt(m_{1}) ≠ wt(m_{2}) , then expressions in both sides of formula (6.3) vanish. For wt(m_{1}) = wt(m_{2}) the straightforward application of the definition of the contravariant form together with Lemma 5.4 gives

(6.4)  C_{μ,Q̂}(m_{1}v_{μ,Q̂} , m_{2} v_{μ,Q})(λ) = S_{μ,Q}(m_{2} , m_{1}v_{μ,Q})(−λ + γ μ + γ ζ′(m_{1})) ,

and using Lemma 6.6 we complete the proof. □

Proof of Theorem 6.4. The first part of the theorem is an easy consequence of the definition of the contravariant form. The second part follows from Proposition 6.7, Lemma 6.6, and the property

(6.5)  C_{μ,Q̂}(f(λ)v , g(λ)ṽ) = f(λ) g(−λ + γ ν) C_{μ,Q̂}(v , ṽ)

for any f, g ∈ Fun(ℂ) , v ∈ Fun(M_{μ,Q̂}[ν]) and ṽ ∈ Fun(M_{μ,Q}[ν]), cf. (6.1). □

Proof of Theorems 6.3 and 6.5. It is clear that Ker π_{μ,Q̂} = {v ∈ Fun(M_{μ,Q̂}) | C_{μ,Q̂}(v , ṽ) = 0 for any ṽ ∈ Fun(M_{μ,Q})}. So Theorem 6.3 is equivalent to claim a) of Theorem 6.5. Claim b) of the latter follows from claim a) and Theorem 6.4.

Since π_{μ,Q̂} v_{μ,Q̂} = ^{∗} v_{μ,Q} ≠ 0 , the subspace Ker π_{μ,Q̂} ⊂ Fun(M_{μ,Q̂}) defines a proper submodule of M_{μ,Q̂}, and by Proposition 5.5 we have that Ker π_{μ,Q̂} ⊂ Ker S_{μ,Q̂}.

Let v ∈ Ker S_{μ,Q̂}. We write it as a linear combination of basis vectors: v = ∑_{m ∈ N} φ_{m}(λ) mv_{μ,Q̂}. Then by Proposition 6.7 for any m̃ ∈ N we have

C_{μ,Q̂}(v , m̃v_{μ,Q})(λ) = ∑_{m ∈ N} φ_{m}(λ)S_{μ,Q̂}(m̃v_{μ,Q} , mv_{μ,Q̂})(λ + γ ζ′′(m̃)) =

= S_{μ,Q̂}(m̃v_{μ,Q} , v)(λ + γ ζ′′(m̃)) = 0 .

Therefore, C_{μ,Q̂}(v , ṽ) = 0 for any ṽ ∈ Fun(M_{μ,Q}) by property (6.5). □

7. Rational dynamical quantum group e_{rat}(𝔰𝔩_{N})

Introduce the spaces Rat(ℂ) , Rat(V) and Rat(Hom(V, W)) similar to the spaces Fun(ℂ) , Fun(V) and Fun(Hom(V, W)), replacing meromorphic function by rational functions in the definitions. Let Rat^{⊗2}(ℂ) = Rat(ℂ) ⊗_{ℂ} Rat(ℂ) .

We define the operator algebra erat^{O}(𝔰𝔩_{N}) and modules over the rational dynamical quantum group e_{rat}(𝔰𝔩_{N}) similar to the elliptic case with the following modification: we replace the spaces of meromorphic functions by the respective spaces of rational functions, substitute the theta function θ(u) by the linear function u ↦ u, and set γ = 1 . For instance, formulae (3.2), (3.4), (4.1) and (3.12) take the form

t_{ab} f(λ^{{1}}, λ^{{2}}) = f(λ^{{1}}− ε_{a} , λ^{{2}}− ε_{b}) t_{ab}

t_{ac} t_{bc} = \frac{λ_{ab}^{\{1\}}+ 1}{λ_{ab}^{\{1\}}− 1} t_{bc} t_{ac} , for a ≠ b,

t̂_{ab} = ∏_{1⩽c<a}λ_{ca}^{{1}} ∏_{1⩽c<b}(λ_{cb}^{{2}})^{−1} t_{ab} , (7.1)

(7.2)  (t_{ab} v)(λ) = ℓ_{ab}(λ)v(λ − ε_{a})  for any v ∈ Rat(V) .

In the last formula V is an e_{rat}(𝔰𝔩_{N})-module. The definitions of highest weight modules, dynamical weights, etc. can be obviously transfered to the rational case.

The rational case can be considered as a degeneration of the elliptic case obtained by rescaling variables: u → γ u, λ → γλ and taking the limit γ → 0 .

Consider the limit R(λ) of the rational version of the R-matrix (1.3) as u → ∞:

(7.3)  R(λ) = ∑_{a,b=1}^{N}E_{aa} ⊗ E_{bb} + ∑_{{a,b=1 \atop a≠b}}^{N} \frac{E_{aa} ⊗ E_{bb} − E_{ab} ⊗ E_{ba}}{λ_{ab}} .

It is a constant solution of the dynamical Yang-Baxter equation:

R^{(12)}(λ − h^{(3)})R^{(13)}(λ)R^{(23)}(λ − h^{(1)}) = R^{(23)}(λ)R^{(13)}(λ − h^{(2)})R^{(12)}(λ) .

R(λ) is the simplest example of the Hecke type dynamical R-matrix, see [EV1].

Let T = ∑_{a,b} E_{ba} ⊗ t_{ab}, where t_{ab} are the generators of erat^{O}(𝔰𝔩_{N}) obeying the rational version of

commutation relations (3.2) – (3.5). Then one can write relations (3.3) – (3.5) in the R-matrix form:

(7.4)  R^{(12)}(λ^{{2}})T^{(13)} T^{(23)} = T^{(23)} T^{(13)} R^{(12)}(λ^{{1}}) .

Let V, W be e_{rat}(𝔰𝔩_{N})-modules. Then the 𝔥-module V ⊗ W is made into an e_{rat}(𝔰𝔩_{N})-module by the rule

(7.5)  ℓ_{ab}(u, λ)|_{V ⊗W} = ∑_{c=1}^{N} ℓ_{cb}(u, λ − h^{(2)}) ⊗ ℓ_{ac}(u, λ) ,

and t_{ab} acts in Rat(V ⊗ W) according to (7.2).

Consider the following element in erat^{O}(𝔰𝔩_{N}) :

(7.6)  t^{∧N} = ∑_{i∈S_{N}}sign(i) t_{N,i_{N}} … t_{1,i_{1}} .

The product D = ∏_{1⩽a<b⩽N}(λ_{ab}^{{1}}/λ_{ab}^{{2}}) t^{∧N} coincides with the top coefficient of the rational version of

the quantum determinant Det T (u) , cf. (3.8). Hence, D is a central element in erat^{O}(𝔰𝔩_{N}) .

Let V be an e_{rat}(𝔰𝔩_{N})-module. The action of D in Rat(V) commutes with multiplication by any function φ(λ) ∈ Rat(ℂ) and, therefore, is given by multiplication by a certain function D(λ) ∈ Rat(End(V)). The module V is called nondegenerate if D(λ) is invertible for generic λ, and semistandard if D(λ) = 1 . For instance, any nondegenerate (standard) highest weight e_{rat}(𝔰𝔩_{N})-module is nondegenerate (semistandard) in the new sense.

One can check that the element D is group-like, it acts in the E_{τ,γ}(𝔰𝔩_{N})-module V ⊗ W by

D(λ)|_{V ⊗W} = D(λ − h^{(2)}) ⊗ D(λ) .

Therefore, a tensor product of nondegenerate (semistandard) e_{τ,γ}(𝔰𝔩_{N})-modules is nondegenerate (semistandard).

Let e_{ab}, a, b = 1, …, N , be the standard generators of the Lie algebra 𝔤𝔩_{N} :

(7.7)  [e_{ab} , e_{cd}] = δ_{bc} e_{ad} − δ_{ad} e_{cb} .

We identify the Lie algebra 𝔰𝔩_{N} with the subalgebra of traceless elements in 𝔤𝔩_{N} :

(7.8)  𝔰𝔩_{N} = {∑_{a,b} x_{ab} e_{ab} | ∑_{a} x_{aa} = 0}

and 𝔥 with the subalgebra of diagonal elements in 𝔰𝔩_{N} : 𝔥 = {∑_{a} x_{aa} e_{aa} | ∑_{a} x_{aa} = 0}. The standard

basis of 𝔥 is h_{a} = e_{aa} − e_{a+1,a+1} , a = 1, …, N − 1 . The assignment e_{ab} ↦ E_{ab}, a, b = 1, …, N , makes ℂ^{N} into the vector representation of 𝔤𝔩_{N} and 𝔰𝔩_{N} .

Let V be an e_{rat}(𝔰𝔩_{N})-module. The elements t̂_{ab} act in Rat(V) as difference operators:

(7.9)  (t̂_{ab} v)(λ) = ℓ̂_{ab}(λ)v(λ − ε_{a}) ,

with coefficients ℓ̂_{ab}(λ) ∈ Rat(End(V)). The module V is called perturbative if these coefficients have the following behaviour as λ goes to infinity in a generic direction:

a) for any a = 1, …, N the function ℓ̂_{aa}(λ) has a limit ℓ̃_{aa} which is an invertible operator;

b) for any a, b = 1, …, N , a ≠ b, the function λ_{ab} ℓ̂_{ab}(λ) has a limit ℓ̃_{ab}.

The operators ℓ̃_{ab}, a, b = 1, …, N satisfy the following commutation relations:

[x, ℓ̃_{bc}] = (x, ε_{b} − ε_{c}) ℓ̃_{bc} , [ℓ̃_{aa} , ℓ̃_{bc}] = 0 ,

for any x ∈ 𝔥 and a, b, c = 1, …, N ,

[ℓ̃_{ab} , ℓ̃_{ba}] = (e_{aa} − e_{bb}) ℓ̃_{aa} ℓ̃_{bb} ,

for a ≠ b, and

[ℓ̃_{ab} , ℓ̃_{bc}] = ℓ̃_{ac} ℓ̃_{bb} ,

for pairwise distinct a, b, c. Hence, the assignment e_{ab} ↦ ℓ̃_{aa}^{−1} ℓ̃_{ab} for a ≠ b, supplemented by the action of 𝔥, makes V into an 𝔰𝔩_{N}-module which we denote by C(V) . We say that V is a perturbation of C(V) . It is clear that V coincide with C(V) as a vector space.

Lemma 7.1. Let V be a perturbative e_{τ,γ}(𝔰𝔩_{N})-module. Then V is nondegenerate.

Proof. The operator D(λ) is invertible for generic λ because it has an invertible limit ℓ̃_{11} … ℓ̃_{NN} as λ goes to infinity in a generic direction. □

Lemma 7.2. A tensor product of perturbative e_{τ,γ}(𝔰𝔩_{N})-modules is perturbative.

Example. The assignment

ℓ̂_{aa}(λ) ↦ 1 − ∑_{a<b⩽N} \frac{E_{bb}}{λ_{ab}^{2}} ,

ℓ̂_{ab}(λ) ↦ \frac{E_{ab}}{λ_{ab}} ,  a ≠ b ,

a, b = 1, …, N , makes ℂ^{N} into an e_{rat}(𝔰𝔩_{N})-module V , which is a perturbation of the vector representation of 𝔰𝔩_{N} . The module V is isomorphic to the vector representation U of e_{rat}(𝔰𝔩_{N}) , cf. (3.13), by the following isomorphism:

φ(λ) = ∑_{a=1}^{N} ∏_{1⩽b<a}λ_{ba} E_{aa} ∈ Mor(V, U) .

Lemma 7.3. Let V be a perturbative e_{rat}(𝔰𝔩_{N})-module. If V has a weight singular vector, then C(V) has a singular vector of the same weight;

Proof. Let v ∈ Rat(V [μ]) be a singular vector. This means that for generic λ the value v(λ) belongs to the subspace K_{λ} = ⋂_{a,b} Ker ℓ̂_{ab}(λ + ε_{a})|_{V [μ]} ⊂ V . It is clear that dim K_{λ} does not depend on λ for

generic λ. Moreover, K_{λ} has a limit K_{∞} as λ goes to infinity in a certain generic direction, and dim K_{∞} = dim K_{λ} ⩾ 1 . To complete the proof we observe that the subspace of singular vectors in C(V)[μ] contains K_{∞}. □

Lemma 7.4. Let V be a perturbative e_{rat}(𝔰𝔩_{N})-module. Then

a) if C(V) is irreducible, then V is irreducible;

b) if C(V) is a highest weight module, then V is a highest weight module of the same highest weight;

c) if C(V) is a Verma module, then V is isomorphic to a Verma module of the same highest weight.

Proof. If V is reducible, then Fun(V) has a nontrivial proper invariant Fun(ℂ)-linear subspace U, which is a direct sum of its weight components. For each weight component U[μ] consider the subspace U_{λ}[μ] ⊂ V spanned by values of functions from U[μ] regular at λ. It is clear that dim_{ℂ} U_{λ}[μ] = dim_{Fun(ℂ)} U[μ] for generic λ. Moreover, U_{λ}[μ] has a limit U_{∞}[μ] as λ goes to infinity in a certain generic direction, dim U_{∞}[μ] = dim U_{λ}[μ] , and the direction can be taken the same for all μ. Then U_{∞} = ⨁_{μ} U_{∞}[μ] is a nontrivial proper invariant subspace in C(V) since dim U_{∞}[μ] < dim V [μ] at least

for some μ. Claim a) is proved.

Let v be the highest weight vector of C(V) . Then the constant function v ∈ Rat(V) is a regular singular vector by the weight reason. Any weight subspace V [μ] has a basis given by vectors of the form e_{a_{1}b_{1}} … e_{a_{k}b_{k}} v. Then the corresponding functions t̂_{a_{1}b_{1}} … t̂_{a_{k}b_{k}} v span Rat(V [μ]), which proves claim b). Claim c) follows from claim b) and comparison of dimensions of weight subspaces. □

Say that an 𝔰𝔩_{N}-module V is admissible if V is a diagonalizable 𝔥-module. Notice, that any highest weight 𝔰𝔩_{N}-module is admissible. In Section 9 we define a functor E from the category of admissible 𝔰𝔩_{N}-modules to the category of semistandard e_{rat}(𝔰𝔩_{N})-modules, cf. Theorem 9.9. We summarize the properties of this functor in the next two theorems.

Theorem 7.5. Let V be an admissible 𝔰𝔩_{N}-module. Then

a) E(V) coincides with V as an 𝔥-module;

b) E(V) is a perturbation of V , moreover, ℓ̃_{aa} = 1 for any a = 1, …, N ;

c) E(V) is a semistandard e_{rat}(𝔰𝔩_{N})-module;

d) if v ∈ V is a weight singular vector, then v ∈ Fun(V) considered as a constant function, is a standard singular vector of the same weight for E(V) ;

e) if V is a highest weight module, then E(V) is a standard highest weight module with the same highest weight and highest weight vector;

f ) if V is a Verma module, then E(V) is isomorpic to the standard Verma module with the same highest weight and highest weight vector.

Theorem 7.6. Let U, V be highest weight 𝔰𝔩_{N}-modules. Then

a) any element of Mor(E(U), E(V)) is a constant function;

b) the map Hom_{𝔰𝔩_{N}}(U, V) → Mor(E(U), E(V)) defined by E is an isomorphism;

c) the above isomorphism coincides with the restriction of the natural embedding of Hom(U, V) into Fun(Hom(U, V)).

The theorems are proved in Section 9.

Let V be a highest weight 𝔰𝔩_{N}-module with highest weight μ and highest weight vector v. Let S be the 𝔰𝔩_{N} Shapovalov form on V , and let S_{μ} be the dynamical Shapovalov pairing for E(V) .

Proposition 7.7. Ker S_{μ} = Fun(Ker S) .

Proof. Since Ker S is an 𝔰𝔩_{N} submodule of V , then E(Ker S) is an e_{rat}(𝔰𝔩_{N}) submodule of E(V) , and Fun(Ker S) ⊂ Ker S_{μ} by Proposition 5.7. On the other hand, V/Ker S is an irreducible 𝔰𝔩_{N}-module, therefore, E(V)/E(Ker S) = E(V/Ker S) is an irreducible e_{rat}(𝔰𝔩_{N})-module by Lemma 7.4. Hence, Ker S_{μ} ⊂ Fun(Ker S) by Proposition 5.7. □

Lemma 7.8. Let a_{i} ≠ b_{i} for any i = 1, …, k and let c_{j} ≠ d_{j} for any j = 1, …, l. Then

(−1)^{k} ∏_{i=1}^{k} λ_{a_{i}b_{i}} ∏_{j=1}^{l} λ_{c_{j}d_{j}} S_{μ}(t̂_{a_{1}b_{1}} … t̂_{a_{k}b_{k}} , t̂_{c_{1}d_{1}} … t̂_{c_{l}d_{l}} v_{μ}) → S(e_{a_{1}b_{1}} … e_{a_{k}b_{k}} v_{μ}, e_{c_{1}d_{1}} … e_{c_{l}d_{l}} v_{μ})

as λ goes to infinity in a generic direction.

8. Finite-dimensional highest weight modules over e_{τ,γ}(𝔰𝔩_{N})

In this section we assume that γ ∉ ℚ + τ ℚ and consider only nondegenerate dynamical weights. We do not mention this assumption explicitly. To save writing we usually formulate the results only for standard highest weight e_{τ,γ}(𝔰𝔩_{N})-modules if they can be generalized to arbitrary highest weight modules by pulling back through automorphisms (3.10).

Let v be a standard singular vector of weight μ, and k be a nonnegative integer.

Lemma 8.1. Let a < b. Then t̂_{ab} t̂_{ba}^{k}v = − \frac{θ((μ_{ab} − k + 1)γ) θ(kγ)}{θ(λ_{ab}) θ(λ_{ab} − (μ_{ab} − 2k + 2)γ)} t̂_{ba}^{k−1}v.

Proof. Take formula (A.11) for a = b, c = d, and replace c by b. Since t̂_{aa}v = t̂_{bb} v = v, we have

t̂_{ab} t̂_{ba}^{k}v = − \frac{θ(λ_{ab}^{\{1\}}− λ_{ab}^{\{2\}}+ (k − 1)γ) θ(kγ)}{θ(λ_{ab}^{\{1\}}) θ(λ_{ab}^{\{2\}})} t̂_{ba}^{k−1}v .

Now apply convention (3.1) and observe that the vector t̂_{ba}^{k−1}v has weight μ − (k − 1)(ε_{a} − ε_{b}) . Since ν_{ab} = (ν, ε_{a} − ε_{b}) for any ν ∈ 𝔥^{∗} and (ε_{a} − ε_{b} , ε_{a} − ε_{b}) = 2 , the lemma is proved. □

Corollary 8.2. Let a < b and μ_{ab} ∉ {0, 1, …, k − 1}. Then t̂_{ba}^{k}v ≠ 0 .

Proof. By Lemma 8.1

t̂_{ab}^{k} t̂_{ba}^{k}v = (−1)^{k} ∏_{j=0}^{k−1} \frac{θ((μ_{ab} − j)γ) θ((j + 1)γ)}{θ(λ_{ab} − jγ) θ(λ_{ab} − (μ_{ab} − j)γ)} v ≠ 0 .

□

Lemma 8.3. t̂_{cd} t̂_{a+1,a}^{k}v = 0 for any c < d, (c, d) ≠ (a, a+1) . t̂_{cc} t̂_{a+1,a}^{k}v = v for any c ≠ a, a+1 .

Lemma 8.4. If (μ, α_{a}) = k − 1 and t̂_{a+1,a}^{k}v ≠ 0 , then ∏_{j=1}^{k} θ(λ_{a,a+1} + jγ) t̂_{a+1,a}^{k}v is a standard singular vector of weight μ − kα_{a}.

Proof. By Lemmas 8.1 and 8.3 the function t̂_{a+1,a}^{k}v is a singular vector and t̂_{cc} t̂_{a+1,a}^{k}v = v for any c ≠ a, a + 1 . On the other hand, it follows from formulae (A.3) and (A.4) that

t̂_{aa} t̂_{a+1,a}^{k}v = \frac{θ(λ_{a,a+1} + kγ)}{θ(λ_{a,a+1})} t̂_{a+1,a}^{k}v , (8.1)

t̂_{a+1,a+1} t̂_{a+1,a}^{k}v = \frac{θ(λ_{a,a+1} − γμ_{a,a+1} + kγ)}{θ(λ_{a,a+1} − γμ_{a,a+1} + 2kγ)} t̂_{a+1,a}^{k}v = \frac{θ(λ_{a,a+1} + γ)}{θ(λ_{a,a+1} + (k + 1)γ)} t̂_{a+1,a}^{k}v (8.2)

because μ_{a,a+1} = (μ, α_{a}) = k − 1 . Hence, multiplying t̂_{a+1,a}^{k}v by the product ∏_{j=1}^{k} θ(λ_{a,a+1} + jγ) one gets a standard singular vector. □

Proposition 8.5. An irreducible standard highest weight e_{τ,γ}(𝔰𝔩_{N})-module with highest weight μ is infinite-dimensional if μ is not a dominant integral weight.

Proof. Let v be the highest weight vector. Assume that μ is not a dominant integral weight, and let a be such that (μ, α_{a}) ∉ ℤ_{⩾0} . Then the functions v , t̂_{a+1,a}v , t̂_{a+1,a}^{2}v , … are linearly independent over Fun(ℂ) because all of them are nonzero by Corollary 8.2 and they have distinct weights with respect to the action of 𝔥. □

Proposition 8.6. Let (μ, α_{a}) = k − 1 . Then t̂_{a+1,a}^{k}v_{μ} generates a submodule of the Verma module M_{μ} isomorphic to the Verma module M_{μ−kα_{a}} .

Proof. By Lemma 8.4 there is a nontrivial morphism φ ∈ Mor(M_{μ−kα_{a}}, M_{μ}) which sends the highest

weight vector v_{μ−kα_{a}} ∈ M_{μ−kα_{a}} to ∏_{j=1}^{k} θ(λ_{a,a+1} + jγ) t̂_{a+1,a}^{k}v_{μ}, and it remains to show that φ is

an embedding. In other words, one has to prove that the induced map Fun(M_{μ−kα_{a}}) → Fun(M_{μ}) is injective.

For any Verma module M_{ν} set Fun_{j}(M_{ν}) = {xv_{ν} | x ∈ eτ,γ^{O}(𝔰𝔩_{N}) , deg(x) ⩽ j} and

Fun_{•}(M_{ν}) = ⨁_{j=0}^{∞} Fun_{j}(M_{ν})/Fun_{j−1}(M_{ν}) .

Let N be given by (4.5). It is clear that the set {xv_{ν} | x ∈ N , deg(x) ⩽ j} is a basis of Fun_{j}(M_{ν}) over Fun(ℂ) , and the set {xv_{ν} | x ∈ N , deg(x) = j} induces a basis of Fun_{j}(M_{ν})/Fun_{j−1}(M_{ν}) . We identify Fun_{•}(M_{ν}) with the space of polynomials in the variables u_{21}, u_{31}, …, u_{N,N−1} with coefficients in Fun(ℂ) : for any monomial t̂_{b_{1}c_{1}} … t̂_{b_{j}c_{j}} ∈ N a class of the function t̂_{b_{1}c_{1}} … t̂_{b_{j}c_{j}} v_{ν} in the quotient Fun_{j}(M_{ν})/Fun_{j−1}(M_{ν}) is mapped to u_{b_{1}c_{1}} … u_{b_{j}c_{j}} .

The map φ : Fun(M_{μ−kα_{a}}) → Fun(M_{μ}) induces the map φ_{•} : Fun_{•}(M_{μ−kα_{a}}) → Fun_{•}(M_{μ}) . Consider φ_{•} as a map from Fun(ℂ)[u_{21}, …, u_{N,N−1}] to itself. By formula (A.10) we find that

φ_{•} (u_{b_{1}c_{1}} … u_{b_{j}c_{j}}) = f(λ − γ ∑_{i=1}^{j}ε_{b_{i}}) u_{b_{1}c_{1}} … u_{b_{j}c_{j}} u_{a+1,a}^{k}

where f(λ) = ∏_{j=1}^{k} θ(λ_{a,a+1} + jγ) . Hence, φ_{•} is injective, and so does φ. □

From now on till the end of the section fix a dominant integral weight μ and set k_{a} = (μ, α_{a}) + 1 , a = 1, …, N − 1 . Notice that k_{a} ∈ ℤ_{>0} for any a. Denote by Z_{μ} the subspace in Fun(M_{μ}) generated over eτ,γ^{O}(𝔰𝔩_{N}) by the functions t̂_{a+1,a}^{k_{a}}v_{μ}, a = 1, …, N − 1 . Notice that the functions t̂_{a+1,a}^{k_{a}}v_{μ} are regular singular vectors, cf. Lemma 8.4.

Let S_{μ} be the Shapovalov pairing for M_{μ}, cf. (5.3).

Proposition 8.7. Z_{μ} ⊂ Ker S_{μ}.

Proof. Lemma 8.4 implies that Z_{μ} is an invariant Fun(ℂ)-linear subspace in Fun(M_{μ}) , therefore it defines a submodule of M_{μ}. Hence, the statement follows from Proposition 5.5. □

Theorem 8.8. Ker S_{μ} = Z_{μ} for generic γ.

Proof. Since both Ker S_{μ} and Z_{μ} are direct sums of their weight components, we have to prove that (Ker S_{μ})[ν] = Z_{μ}[ν] for any ν ⩽ μ. Notice that (Ker S_{μ})[μ] = Fun(ℂ)v_{μ} = Z_{μ}[μ] .

By Proposition 8.7 it suffices to prove that (Ker S_{μ})[ν] and Z_{μ}[ν] have the same dimensions. For doing this we employ the deformation arguments.

Consider the Verma module M_{μ} M^{𝔰𝔩} of highest weight μ over 𝔰𝔩_{N} and the Shapovalov form S_{μ} S^{𝔰𝔩} on it. Let M_{μ} M^{rat} = E(M_{μ} M^{𝔰𝔩}) , see Theorem 7.5, and let S_{μ} S^{rat} be the corresponding dynamical Shapovalov pairing. Recall that, M_{μ} M^{rat} is isomorphic to the Verma module of highest weight μ over e_{rat}(𝔰𝔩_{N}) . By abuse of notation we denote generators of eτ,γ^{O}(𝔰𝔩_{N}) and erat^{O}(𝔰𝔩_{N}) by the same letters and write v_{μ} for the highest weight vector of each of the modules M_{μ}, M_{μ} M^{𝔰𝔩} and M_{μ} M^{rat} .

Let Z_{μ} Z^{𝔰𝔩} be the 𝔰𝔩_{N} submodule in M_{μ} M^{𝔰𝔩} generated by singular vectors e_{a+1,a}^{k_{a}}v_{μ}, a = 1, …, N − 1 , and let Z_{μ} Z^{rat} be the subspace in Rat(M_{μ} M^{rat}) generated over erat^{O}(𝔰𝔩_{N}) by functions t̂_{a+1,a}^{k_{a}}v_{μ}, a = 1, …, N − 1 . It is known that Ker S_{μ} S^{𝔰𝔩} = Z_{μ} Z^{𝔰𝔩}. By Theorem 7.5 and Proposition 7.7 we have that Ker S_{μ} S^{rat} = Rat(S_{μ} S^{𝔰𝔩}) .

Lemma 8.9. ∏_{j=1}^{k_{a}}(λ_{a,a+1} + j) t̂_{a+1,a}^{k_{a}}v_{μ} = (−1)^{k_{a}} e_{a+1,a}^{k_{a}}v_{μ} for any a = 1, …, N − 1 .

Proof. Both ∏_{j=1}^{k_{a}}(λ_{a,a+1} + j) t̂_{a+1,a}^{k_{a}}v_{μ} and e_{a+1,a}^{k_{a}}v_{μ} have weight μ − k_{a}α_{a}. So they are proportional

over Rat(ℂ) because dim_{ℂ} M_{μ} M^{𝔰𝔩}[μ − k_{a}α_{a}] = 1 . The proportionality coefficient is a constant function since both of them are standard singular vectors by Theorem 7.5 and the rational version of Lemma 8.4. The constant equals 1 because λ_{a,a+1}^{k_{a}}t̂_{a+1,a}^{k_{a}}v_{μ} → (−1)^{k_{a}} e_{a+1,a}^{k_{a}}v_{μ} as λ goes to infinity in a generic direction. □

Corollary 8.10. Ker S_{μ} S^{rat} = Z_{μ} Z^{rat} = Rat(Z_{μ} Z^{𝔰𝔩}) .

Since the elliptic case is a deformation of the rational one we obtain that for generic γ

(8.3)  dim_{Fun(ℂ)}(Ker S_{μ})[ν] ⩽ dim_{Rat(ℂ)}(Ker S_{μ} S^{rat})[ν] = dim_{ℂ} (Ker S_{μ} S^{𝔰𝔩})[ν] =
= dim_{ℂ} Z_{μ} Z^{𝔰𝔩}[ν] = dim_{Rat(ℂ)} Z_{μ} Z^{rat}[ν] ⩽ dim_{Fun(ℂ)} Z_{μ}[ν] ⩽ dim_{Fun(ℂ)}(Ker S_{μ})[ν] .

Here the last inequality is due to Proposition 8.7. Therefore, all the dimensions in (8.3) are the same, which proves the theorem. The rest of the proof is a justification of this informal reasoning.

For a pair of sequences a = (a_{1}, …, a_{j}) and b = (b_{1}, …, b_{j}) let t̂_{ab} = t̂_{a_{1}b_{1}} … t̂_{a_{j}b_{j}} . Set

wt(a,b) = wt(t̂_{ab}) = ∑_{i=1}^{j}(ε_{a_{i}} − ε_{b_{i}}) , ζ(a,b) = ∑_{i=1}^{j} ε_{b_{i}} .

Let M = {(a,b) | t̂_{ab} ∈ N } where the set N is defined in (4.5). Set

M[β] = {m ∈ M | wt(m) = β} , M_{μ}[β] = ⋃_{a=1}^{N−1} M[β + k_{a}α_{a}] .

Let β = ν − μ. Introduce a matrix A^{γ}(λ) with entires labeled by pairs of elements of M[β] :

(8.4)  A_{m,m′}^{γ}(λ) = S_{μ}(t̂_{m} , t̂_{m′} v_{μ})(λ + γ ζ(m)) ,

and a matrix B^{γ}(λ) with entries labeled by pairs (m, m′) ∈ M[β] × M_{μ}[β] and given by the rule:

m′ t̂_{a+1,a}^{k_{a}}v_{μ} = ∑_{m ∈ M[β]}B_{m,m′}^{γ}(λ) t̂_{m}v_{μ} , m′ ∈ M[β + k_{a}α_{a}] .

Both A^{γ}(λ) and B^{γ}(λ) are meromorphic functions of γ , λ. Moreover, if γ → 0 , then

(8.5)  A^{γ}(γλ) → A^{rat}(λ) ,  B^{γ}(γλ) → B^{rat}(λ) ,

where A^{rat}(λ) and B^{rat}(λ) are defined in the same way for the rational case (in formula (8.4) the argument in the right hand side is λ + ζ(m) for the rational case).

Each matrix naturally defines a linear map: A^{γ}(λ) and A^{rat}(λ) act in M_{ℂ}[β] = ⨁_{m ∈ M[β]}ℂm, whilst

B^{γ}(λ) and B^{rat}(λ) map ⨁_{m ∈ M_{μ}[β]}ℂm to M_{ℂ}[β] .

Given functions φ_{m}(λ) ∈ Fun(ℂ) , m ∈ M[β] , consider a vector φ̌(λ) = ∑_{m ∈ M[β]}φ_{m}(λ) m ∈ M_{ℂ}[β] .

Set φ⃗(λ) = ∑_{m ∈ M[β]}φ_{m}(λ) t̂_{m}v_{μ} ∈ Fun(M_{μ}) .

Lemma 8.11.

a) φ⃗ ∈ (Ker S_{μ})[ν] iff φ̌(λ) ∈ Ker A^{γ}(λ) for generic λ;

b) φ⃗ ∈ (Ker S_{μ} S^{rat})[ν] iff φ̌(λ) ∈ Ker A^{rat}(λ) for generic λ;

c) φ⃗ ∈ Z_{μ}[ν] iff φ̌(λ) ∈ Im B^{γ}(λ) for generic λ;

d) φ⃗ ∈ Z_{μ} Z^{rat}[ν] iff φ̌(λ) ∈ Im B^{rat}(λ) for generic λ.

Proof. Claims a) and b) follow from formulae (5.2) – (5.4) and their rational versions, respectively. Claims c) and d) are straightforward. □

Corollary 8.10 gives that Ker A^{rat}(λ) = Im B^{rat}(λ) for generic λ. The standard deformation reasoning, cf. (8.5), shows that

(8.6) dim_{ℂ} Ker A^{γ}(λ) ⩽ dim_{ℂ} Ker A^{rat}(λ) = dim_{ℂ} Im B^{rat}(λ) = dim_{ℂ} Im B^{γ}(λ) ⩽ dim_{ℂ} Ker A^{γ}(λ)

for generic λ, if γ is generic. Here the last inequality is due to Proposition 8.7 which implies that Im B^{γ}(λ) ⊂ Ker A^{γ}(λ) for generic λ. Therefore, all the dimensions in (8.6) are the same. Hence Ker A^{γ}(λ) = Im B^{γ}(λ) for generic λ, if γ is generic. By Lemma 8.11 we get that (Ker S_{μ})[ν] = Z_{μ}[ν] for generic γ. Theorem 8.8 is proved. □

Let V_{μ} be the irreducible standard highest weight e_{τ,γ}(𝔰𝔩_{N})-module of highest weight μ. Let N_{μ} be the e_{rat}(𝔰𝔩_{N})-submodule of M_{μ} such that Fun(N_{μ}) = Ker S_{μ}. We have V_{μ} = M_{μ}/N_{μ}, cf. Corollary 5.6. Let V_{μ} V ^{𝔰𝔩} be the irreducible 𝔰𝔩_{N}-module of highest weight μ. Set d_{μ}[ν] = dim_{ℂ} V_{μ} V ^{𝔰𝔩}[ν] .

Proof of Theorem 5.8. It follows from the proof of Theorem 8.8 that for generic γ

(8.7)  dim_{ℂ} V_{μ}[ν] = dim_{ℂ} M_{μ}[ν] − dim_{Fun(ℂ)}(Ker S_{μ})[ν] =
= dim_{ℂ} M_{μ} M^{𝔰𝔩}[ν] − dim_{ℂ} (Ker S_{μ} S^{𝔰𝔩})[ν] = dim_{ℂ} V_{μ} V ^{𝔰𝔩}[ν] = d_{μ}[ν] .

Since dim_{Fun(ℂ)}(Ker S_{μ})[ν] can only jump up at specific γ, we have that dim_{ℂ} V_{μ}[ν] ⩽ d_{μ}[ν] for arbitrary γ. □

Proof of Theorem 5.9. We have proved already that dim_{ℂ} V_{μ}[ν] = d_{μ}[ν] for any ν, if γ is generic, cf. (8.7). Consider an e_{τ,γ}(𝔰𝔩_{N})-module U_{μ} defined in the following way. Let U_{μ} = V_{μ} if γ is generic, that is, if dim_{ℂ} V_{μ}[ν] = d_{μ}[ν] . Otherwise, define U_{μ} by analytic continuation from generic γ. It is not difficult to justify the given definition of U_{μ}, and to see that U_{μ} is a highest weight e_{τ,γ}(𝔰𝔩_{N})-module with highest weight μ and dim_{ℂ} U_{μ}[ν] = d_{μ}[ν] for any ν.

If μ is a dominant integral weight, then w ·μ < μ and d_{μ}[w ·μ] = 0 for any nontrivial element w ∈ W. Hence, U_{μ} is irreducible by Corollary 4.4 if γ ∉ ℚ + τ ℚ, that is, U_{μ} = V_{μ}. The theorem is proved. □

Remark. There is another approach to constructing finite-dimensional irreducible representation of e_{τ,γ}(𝔰𝔩_{N}) . One can start from the vector representation of e_{τ,γ}(𝔰𝔩_{N}) and apply the fusion procedure technique developed in the nondynamical case, cf. [C], [N]. If γ ∉ ℚ + τ ℚ, then any irreducible finitedimensional standard highest weight e_{τ,γ}(𝔰𝔩_{N})-module can be obtained in this way. The symmetric and exterior powers of the vector representation of e_{τ,γ}(𝔰𝔩_{N}) has been constructed by this technique in [FV2]. We will address this approach elsewhere.

9. Definition of the functor E

In this section we construct a functor from the category of admissible 𝔰𝔩_{N}-modules to the category of semistandard e_{rat}(𝔰𝔩_{N})-modules. The construction is similar to the construction of the functor from the category of finite-dimensional 𝔰𝔩_{N}-modules to the category of rational representations of the exchange quantum group F (SL(N)), developed in [EV2]. We discuss in detail the relation of two constructions in Section 10.

Let 𝔫_{+} = { ∑_{a<b} x_{ab} e_{ab} } and 𝔫_{−} = { ∑_{a>b} x_{ab} e_{ab} } be the standard nilpotent subalgebras in 𝔰𝔩_{N} , and

let 𝔟_{±} = 𝔥 ⊕ 𝔫_{±}. Set

Ξ = \frac{1}{2} ∑_{a=1}^{N} e_{aa}^{2} − \frac{1}{2N} ( ∑_{a=1}^{N} e_{aa})^{2} ∈ U(𝔥) .

Proposition 9.1. There exists a unique power series J (λ; z) in z with coefficients in U(𝔰𝔩_{N}) ⊗ U(𝔰𝔩_{N}) valued functions of λ with the properties:

a) J (λ; z) satisfies the rational ABRR equation

(9.1)  J (λ; z) (1 ⊗ (λ − z Ξ)) = (1 ⊗ (λ − z Ξ) + z ∑_{1⩽a<b⩽N}e_{ab} ⊗ e_{ba}) J (λ; z) ;

b) the coefficients of the series (J (λ; z) − 1) are U(𝔟_{+})𝔫_{+} ⊗ 𝔫_{−}U(𝔟_{−})-valued functions of λ.

Proof. Let J (λ; z) = ∑_{k=0}^{∞} J_{k}(λ)z^{k}. Equation (9.1) is equivalent to certain recurrence relations for coefficients J_{k}(λ) with the initial condition J_{0}(λ) = 1 . It is straightforward to verify that at each step the recurrence relations uniquely determine J_{k} from J_{0}, …, J_{k−1} . □

It follows from the proof of the last proposition that the coefficients of the series J (λ; z) are rational functions of λ, and for any x ∈ 𝔥

(9.2)  [J (λ; z), x ⊗ 1 + 1 ⊗ x] = 0 .

Denote by Δ : U(𝔰𝔩_{N}) → U(𝔰𝔩_{N}) ⊗ U(𝔰𝔩_{N}) the coproduct for U(𝔰𝔩_{N}) . Theorem 9.2. The series J (λ; z) satisfies the equation

(9.3)  J^{((12)3)}(λ; z)J^{(12)}(λ − zh^{(3)}; z) = J^{(1(23))}(λ; z)J^{(23)}(λ; z) .

Here J^{((12)3)} = (Δ⊗id)(J) , J^{(1(23))} = (id⊗Δ)(J) , J^{(12)} = J ⊗1 , J^{(23)} = 1⊗J , and the meaning of J^{(12)}(λ − zh^{(3)}; z) is explained below, cf. (9.4). Proof. The statement is a degeneration of Theorem 3.1 in [ESS], and can be proved in the same way. □ Remark. Let x_{1}, …, x_{N−1} be a basis of 𝔥^{∗}, and let x^{1}, …, x^{N−1} be the dual basis of 𝔥. Write λ = λ^{1}x_{1} + … + λ^{N−1}x_{N−1} . For a rational function f(λ) we define a series f(λ − zh^{(3)}) by the Taylor series expansion:

(9.4)  f(λ − zh^{(3)}) = f(λ; z) − z ∑_{a=1}^{N−1} \frac{∂f(λ)}{∂λ^{a}} (x^{a})^{(3)} + …,

and extend the definition to series in z with coefiicients in rational functions of λ in the natural way. Remark. The equation (9.3) is usually called the dynamical 2-cocycle condition.

Define a rational exchange matrix R(λ; z) by the rule:

(9.5)  R(λ; z) = (J (λ; z))^{−1}J^{(21)}(λ; z) .

Theorem 9.3. R(λ) satisfies the dynamical Yang-Baxter equation:

R^{(12)}(λ − zh^{(3)}; z)R^{(13)}(λ; z)R^{(23)}(λ − zh^{(1)}; z) = R^{(23)}(λ; z)R^{(13)}(λ − zh^{(2)}; z)R^{(12)}(λ; z) .

The statement follows from Theorem 9.2 and cocommutativity of the coproduct Δ .

Say that an n-tuple V_{1}, …, V_{n} of 𝔰𝔩_{N}-modules is admissible if for any pairwise distinct i_{1}, …, i_{k} the tensor product V_{i_{1}} ⊗ … ⊗ V_{i_{k}} is a diagonalizable 𝔥-module; that is, V_{1}, …, V_{n} are diagonalizable 𝔥-modules and all weight subspaces of any tensor product V_{i_{1}} ⊗ … ⊗ V_{i_{k}} are finite-dimensional.

Let V, W be an admissible pair of 𝔰𝔩_{N}-modules. Denote by J_{V W} (λ; z) ∈ End(V ⊗ W) the action of J (λ; z) in the tensor product V ⊗ W. It follows from the explicit form of recurrence relations in the proof of Proposition 9.1 that there is a unique function J_{V W} (λ) ∈ Rat(End(V ⊗ W)) such that the series J_{V W} (λ; z) coincides with the expansion of J_{V W} (λ/z) at z = 0 . The function J_{V W} (λ) admits the following description. Lemma 9.4. J_{V W} (λ) is the unique solution of the equation

J_{V W} (λ) (1 ⊗ (λ − Ξ))|_{V ⊗W} = (1 ⊗ (λ − Ξ) + ∑_{1⩽a<b⩽N}e_{ab} ⊗ e_{ba})|_{V ⊗W} J_{V W} (λ) ,

such that (J_{V W} (λ) − 1) ∈ (U(𝔟_{+})𝔫_{+} ⊗ 𝔫_{−}U(𝔟_{−}))|_{V ⊗W} . Moreover, J_{V W} (λ) commutes with the action of 𝔥 in V ⊗ W, cf. (9.2). Proposition 9.5. For any admissible triple of 𝔰𝔩_{N}-modules U, V, W we have

(9.6)  J_{U⊗V,W} (λ)(J_{UV} (λ − h^{(3)}) ⊗ 1) = J_{U,V ⊗W} (λ)(1 ⊗ J_{V W} (λ)) .

Proof. The function (J_{UV} (λ − h^{(3)}) ⊗ 1) is defined by the rule (1.1). To get relation (9.6) from formula (9.3) one needs to verify that the series obtained by expansion of (J_{UV} (z^{−1}λ − h^{(3)}) ⊗ 1) at z = 0 coincides with the action of J^{(12)}(λ − zh^{(3)}; z) , defined by (9.4), in U ⊗ V ⊗ W, which is simple. □

For any A ∈ End(W ⊗ V) let A^{(21)} = P A P^{−1} ∈ End(V ⊗ W) where P : W ⊗ V → V ⊗ W is the permutation map: P (x ⊗ y) = y ⊗ x. Set

(9.7)  R_{V W} (λ) = (J_{V W} (λ))^{−1}(J_{W V} (λ))^{(21)} .

It is clear that the action of the series R(λ; z) in V ⊗ W coincides with the expansion of R_{V W} (λ/z) at z = 0 . Like in the proof of Proposition 9.5 we get the following assertion from Theorem 9.3.

Proposition 9.6. For any admissible triple of 𝔰𝔩_{N}-modules U, V, W we have

R_{UV}^{(12)}(λ − h^{(3)})R_{UW}^{(13)}(λ)R_{V W}^{(23)}(λ − h^{(1)}) = R_{V W}^{(23)}(λ)R_{UW}^{(13)}(λ − h^{(2)})R_{UV}^{(12)}(λ) .

Let J_{V W} (λ) be the fusion matrix for U(𝔰𝔩_{N}) defined in [EV2], and let

(9.8)  R_{V W} (λ) = (J_{V W} (λ))^{−1}(J_{W V} (λ))^{(21)}

be the corresponding dynamical R-matrix. Let X be the matrix ∑_{a=1}^{N}(−1)^{a−1}E_{a,N−a+1} considered as

an element of the group SL(N) , and let w_{X} be the longest element of the Weil group. We have Ad_{X}(e_{ab}) = e_{N−a+1,N−b+1} and w_{X}(ε_{a}) = ε_{N−a+1} for any a, b = 1, …, N .

Lemma 9.7. For any finite-dimensional 𝔰𝔩_{N}-modules V, W we have

(9.9)  J_{V W} (λ) = (X ⊗ X)J_{V W} (w_{X}(λ + ρ))(X ⊗ X)^{−1}.

Proof. It is shown in [EV2, Section 9] that J_{V W} (λ) is the unique solution of the equation

J_{V W} (λ) (1 ⊗ (λ + ρ − Ξ))|_{V ⊗W} = (1 ⊗ (λ + ρ − Ξ) + ∑_{1⩽a<b⩽N}e_{ba} ⊗ e_{ab})|_{V ⊗W} J_{V W} (λ) ,

such that (J_{V W} (λ) − 1) ∈ (𝔫_{−}U(𝔟_{−}) ⊗ U(𝔟_{+})𝔫_{+})|_{V ⊗W} . By Lemma 9.4 the right hand side of formula

(9.9) has the same properties, which proves the claim. □

Corollary 9.8. R_{V W} (λ) = (X ⊗ X)R_{V W} (w_{X}(λ + ρ))(X ⊗ X)^{−1}.

Henceforward, let U be the vector representation of 𝔰𝔩_{N} . By formula (36) in [EV2] we have

(9.10)  R_{UU}(λ) = ∑_{a,b=1}^{N}E_{aa} ⊗ E_{bb} − ∑_{1⩽a<b⩽N} \frac{E_{bb} ⊗ E_{aa}}{(λ_{ab} − a + b)^{2}} − ∑_{{a,b=1 \atop a≠b}}^{N} \frac{E_{ab} ⊗ E_{ba}}{λ_{ab} − a + b} ,

Therefore,

(9.11)  R_{UU}(λ) = ∑_{a,b=1}^{N}E_{aa} ⊗ E_{bb} − ∑_{1⩽a<b⩽N} \frac{E_{aa} ⊗ E_{bb}}{λ_{ab}^{2}} − ∑_{{a,b=1 \atop a≠b}}^{N} \frac{E_{ab} ⊗ E_{ba}}{λ_{ab}} .

Let V be an admissible 𝔰𝔩_{N}-module. Introduce functions ^{V} ℓ̂_{ab} ∈ Rat(End(V)), a, b = 1, …, N , by the equality

R_{UV} (λ) = ∑_{a,b=1}^{N}E_{ba} ⊗ ^{V} ℓ̂_{ab}(λ) .

Theorem 9.9. Let V be an admissible 𝔰𝔩_{N}-module. Then the rule (t̂_{ab} v)(λ) = ^{V} ℓ̂_{ab}(λ)v(λ − ε_{a}) , for any a, b = 1, …, N and any v ∈ Rat(V) , endows V with an e_{rat}(𝔰𝔩_{N})-module structure. The constructed e_{rat}(𝔰𝔩_{N})-module is denoted by E(V) .

Proof. The statement follows from Proposition 9.6, and formulae (7.1), (7.4), (7.9) and (9.11). □

We define the functor E from the category of admissible 𝔰𝔩_{N}-modules to the category of semistandard e_{rat}(𝔰𝔩_{N})-modules sending an object V to E(V) and a morphism φ ∈ Hom(V, W) to the corresponding constant function φ ∈ Mor(V, W) .

Proof of Theorem 7.5. Claim a) of the theorem is immediate. Say that f(λ) = O(|λ|^{k}) if f(sλ) = O(s^{k}) as s → +∞ for generic λ. From Lemma 9.4 and formula (9.7) we have that

J_{UV} (λ) = 1 − ∑_{1⩽a<b⩽N}\frac{E_{ba} ⊗ e_{ab}}{λ_{ab}} + O(|λ|^{−2}) ,

R_{UV} (λ) = 1 + ∑_{{a,b=1 \atop a≠b}}^{N} \frac{E_{ba} ⊗ e_{ab}}{λ_{ab}} + O(|λ|^{−2}) ,

which proves claim b). Claim c) follows from Lemma 7.1. Claims e) and f) follow from claims b) and d), and Lemma 7.4.

To prove claim d) one should show that for any singular vector v ∈ V we have ^{V} ℓ̂_{aa}(λ) v = v for any a, and ^{V} ℓ̂_{ab}(λ) v = 0 for any a < b. By Lemma 9.4 and formula (9.7) we see that ( ^{V} ℓ̂_{ab}(λ) − 1) ∈ (𝔫_{−}U(𝔰𝔩_{N})𝔫_{+})|_{V} for any a, and ^{V} ℓ̂_{ab}(λ) ∈ (U(𝔰𝔩_{N})𝔫_{+})|_{V} for any a < b, which implies claim d).

It remains to prove claim c. The element D acts in Rat(V) as multiplication by D(λ) . It is clear from the definition of E(V) that there exists some independent of V element in a certain completion of U(𝔰𝔩_{N}) such that its action in V coincides with D(λ) . Since an element of U(𝔰𝔩_{N}) is uniquely determined by its action in highest weight modules, it suffices to prove claim c) under the assumption that V is a highest weight module. In the last case claim c) follows from claim d). Theorem 7.5 is proved. □

The proof of Theorem 7.6 is similar to the proof of Theorem 45 in [EV2].

10. Exchange quantum group F (SL(N))

For any a = 1, …, N let i^{a} = (1, …, a − 1, a + 1, …, N) . Set

t̄_{ab} = (−1)^{a+b} ∑_{j∈S_{N−1}}sign(j) t_{i_{N−1}^{a}, i_{j_{N−1}}^{b}} … t_{i_{1}^{a}, i_{j_{1}}^{b}} .

Lemma 10.1. ∑_{c=1}^{N} t̄_{ac} t_{bc} = δ_{ab} t^{∧N}, where t^{∧N} is defined by (7.6).

Proof. The formula coincides with the equality of the top coefficients in the rational version of formula (B.6) for k = 1 and T (u) = T (u) , cf. (3.6). □

Consider the exchange quantum group F (SL(N)) defined in [EV2]. It admits the following description, see [EV2, Section 5.3]. F (SL(N)) is a unital associative algebra over ℂ generated by functions f ∈ Rat^{⊗2}(ℂ) and elements Tab^{+} , Tab^{−}, a, b = 1, …, N , subject to relations (10.1) – (10.3) and (10.5).

Let R(λ) = R_{UU}(λ) , cf. (9.10). Set T^{±} = ∑_{a,b=1}^{N} E_{ab} ⊗ Tab^{±}. The defining relations for F (SL(N)) are

T^{+} T^{−} = T^{−} T^{+} = id ⊗ 1 , (10.1)

Tab^{+} f(λ^{{1}}, λ^{{2}}) = f(λ^{{1}}− ε_{b} , λ^{{2}}− ε_{a}) Tab^{+} (10.2)

for any f ∈ Rat^{⊗2}(ℂ) ,

(10.3)  R^{(12)}(λ^{{2}})T^{(13)}T^{(23)} = T^{(23)}T^{(13)} R^{(12)}(λ^{{1}})

where T^{(13)} = ∑_{a,b} E_{ab} ⊗ id ⊗ Tab^{+} and T^{(23)} = ∑_{a,b} id ⊗ E_{ab} ⊗ Tab^{+} , and relation (10.5).

Remark. In this paper the variables λ^{{1}}, λ^{{2}} and the generators Tab^{+}, Tab^{−} correspond to the variables λ^{2}, λ^{1} and the generators L_{ab}, L_{ab}^{−1} in [EV2].

For any permutation i ∈ S_{N} let Λ_{i}(λ) = ∏_{{1⩽a<b⩽N \atop i_{a}<i_{b}}}(1 + λ_{ab}^{−1}) . Set

(10.4)  Det T^{+} = \frac{1}{Λ_{id}(λ^{\{1\}})} ∑_{i∈S_{N}}sign(i) Λ_{i}(λ^{\{2\}}) Ti_{N},N^{+} … Ti_{1},1^{+}

where id = (1, …, N) . The last defining relation for F (SL(N)) is

(10.5)  Det T^{+} = 1 .

Remark. The element Det T^{+} corresponds to the element D in [EV2]. Formula (10.4) can be derived from the definition of D therein. The complete proof of formula (10.4) will appear elsewhere.

Recall that given a diagonalizable 𝔥-module V we assume the following action of Rat^{⊗2}(ℂ) on V-valued functions:

f : v(λ) ↦ f(λ, λ − μ) v(λ)

for any f ∈ Rat^{⊗2}(ℂ) and any function v(λ) with values in V [μ] .

A rational dynamical representation of F (SL(N)) is a diagonalizable 𝔥-module V endowed with an action of F (SL(N)) on V-valued meromorphic functions by difference operators with rational coefficients:

(10.6)  (Tab^{+} v)(λ) = L_{ab}(λ)v(λ − ε_{b}) ,  a, b = 1, …, N ,

where L_{ab}(λ) ∈ Rat(End(V)) are suitable functions.

Proposition 10.2. Let V be a rational dynamical representation of F (SL(N)). Then the rule

(10.7)  (t_{ab} v)(λ) = ∏_{1⩽c<a}(λ_{ca} + 1)^{−1} ∏_{1⩽c<b}(λ_{cb} + e_{bb} − e_{cc} + 1) L_{ba}(λ − ρ)v(λ − ε_{a})

for any a, b = 1, …, N and any v ∈ Rat(V) , endows V with a structure of a semistandard e_{rat}(𝔰𝔩_{N}) module.

The proof is straightforward.

Proposition 10.3. Let V be a semistandard e_{rat}(𝔰𝔩_{N})-module. Then formulae (10.6), (10.7) make V into a rational dynamical representation of F (SL(N)).

Proof. It is straightforward to verify relations (10.2), (10.3) and (10.5). To complete the proof it remains to find the action of elements Tab^{−} to obey relations (10.1). This can be done using Proposition 10.1, since the e_{rat}(𝔰𝔩_{N})-module V is nondegenerate. □

The last two propositions define a functor F from the category of rational dynamical representations of F (SL(N)) to the category of semistandard e_{rat}(𝔰𝔩_{N})-modules: an object V goes to itself and a morphism φ(λ) ∈ Rat(Hom(V, W)) goes to φ(λ − ρ) ∈ Mor(V, W) . Furthermore, the propositions imply the following assertion.

Theorem 10.4. The functor F is an equivalence of the categories.

For both categories involved in the last theorem the subcategories of finite-dimensional objects are tensor categories, the tensor product of rational dynamical representations of F (SL(N)) being defined in [EV2]. One can show that the restriction of the functor F to these subcategories is an equivalence of tensor categories.

Let G be the functor from the category of finite-dimensional 𝔰𝔩_{N}-modules to the category of finitedimensional dynamical representations of F (SL(N)) defined in [EV2]: an 𝔰𝔩_{N}-module V goes to the representation G(V) of F (SL(N)) given by the rule

(10.8)  R_{UV} (λ) = ∑_{a,b=1}^{N}E_{ab} ⊗ L_{ab}(λ) ,

and a morphism φ ∈ Hom(V, W) goes to the corresponding constant function φ ∈ Rat(Hom(V, W)). The composition Ẽ = F ∘ G is a functor from the category of finite-dimensional 𝔰𝔩_{N}-modules to the category of semistandard e_{rat}(𝔰𝔩_{N})-modules.

Theorem 10.5. The functor Ẽ is isomorphic to the restriction of the functor E to the category of finite-dimensional 𝔰𝔩_{N}-modules.

The theorem is proved in Appendix F.

Remark. Let V be an irreducible finite-dimensional 𝔰𝔩_{N}-module. Then the e_{rat}(𝔰𝔩_{N})-module E(V) is an irreducible standard highest weight module over the dynamical quantum group e_{rat}(𝔰𝔩_{N}) . Such modules have been described in Section 8. Applying the functor inverse to F we get a new description of the dynamical representations of F (SL(N)) induced from irreducible finite-dimensional 𝔰𝔩_{N}-modules. This new description is a new highest weight module theory for the exchange dynamical quantum group F (SL(N)).

Appendix A. Commutation relations in eτ,γ^{O}(𝔰𝔩_{N})

In this section we collect useful commutation relations which hold in the operator algebra eτ,γ^{O}(𝔰𝔩_{N}) .

In the definition of eτ,γ^{O}(𝔰𝔩_{N}) one can replace relations (3.5) by the following relations:

(A.1)  \frac{θ(λ_{ac}^{\{1\}}− γ)}{θ(λ_{ac}^{\{1\}})} t_{ab} t_{cd} − \frac{θ(λ_{bd}^{\{2\}}− γ)}{θ(λ_{bd}^{\{2\}})} t_{cd} t_{ab} = \frac{θ(λ_{ac}^{\{1\}}+ λ_{bd}^{\{2\}}) θ(γ)}{θ(λ_{ac}^{\{1\}}) θ(λ_{bd}^{\{2\}})} t_{cb} t_{ad} ,

for a ≠ c and b ≠ d. The last formula implies that

(A.2)  t̂_{aa} t̂_{bb} − t̂_{bb} t̂_{aa} = \frac{θ(λ_{ab}^{\{1\}}+ λ_{ab}^{\{2\}}) θ(γ)}{θ(λ_{ab}^{\{1\}}) θ(λ_{ab}^{\{2\}})} t̂_{ba} t̂_{ab}

for a < b. Under the same assumption we have

\frac{θ(λ_{ab}^{\{2\}}− γ) θ(λ_{ab}^{\{2\}}+ γ)}{θ(λ_{ab}^{\{2\}})^{2}} t̂_{aa} t̂_{bb} − \frac{θ(λ_{ab}^{\{1\}}− γ) θ(λ_{ab}^{\{1\}}+ γ)}{θ(λ_{ab}^{\{1\}})^{2}} t̂_{bb} t̂_{aa} = \frac{θ(λ_{ab}^{\{1\}}+ λ_{ab}^{\{2\}}) θ(γ)}{θ(λ_{ab}^{\{1\}}) θ(λ_{ab}^{\{2\}})} t̂_{ab} t̂_{ba} ,

t̂_{ab} t̂_{ba} − \frac{θ(λ_{ab}^{\{1\}}− γ) θ(λ_{ab}^{\{1\}}+ γ)}{θ(λ_{ab}^{\{1\}})^{2}} t̂_{ba} t̂_{ab} = − \frac{θ(λ_{ab}^{\{1\}}− λ_{ab}^{\{2\}}) θ(γ)}{θ(λ_{ab}^{\{1\}}) θ(λ_{ab}^{\{2\}})} t̂_{aa} t̂_{bb} ,

t̂_{ab} t̂_{ba} − \frac{θ(λ_{ab}^{\{2\}}− γ) θ(λ_{ab}^{\{2\}}+ γ)}{θ(λ_{ab}^{\{2\}})^{2}} t̂_{ba} t̂_{ab} = − \frac{θ(λ_{ab}^{\{1\}}− λ_{ab}^{\{2\}}) θ(γ)}{θ(λ_{ab}^{\{1\}}) θ(λ_{ab}^{\{2\}})} t̂_{bb} t̂_{aa} .

More general relations are listed below. We assume that a < c and b < d therein.

(A.3)  t̂_{ab} t̂_{cb}^{k} = \frac{θ(λ_{ac}^{\{1\}}+ kγ)}{θ(λ_{ac}^{\{1\}})} t̂_{cb}^{k} t̂_{ab} ,  t̂_{ab}^{k} t̂_{cb} = \frac{θ(λ_{ac}^{\{1\}}+ γ)}{θ(λ_{ac}^{\{1\}}− (k − 1)γ)} t̂_{cb} t̂_{ab}^{k} ,

(A.4)  t̂_{ad} t̂_{ab}^{k} = \frac{θ(λ_{bd}^{\{2\}}− kγ)}{θ(λ_{bd}^{\{2\}})} t̂_{ab}^{k} t̂_{ad} ,  t̂_{ad}^{k} t̂_{ab} = \frac{θ(λ_{bd}^{\{2\}}− γ)}{θ(λ_{bd}^{\{2\}}+ (k − 1)γ)} t̂_{ab} t̂_{ad}^{k} ,

(A.5)  t̂_{ab} t̂_{cd} − t̂_{cd} t̂_{ab} = \frac{θ(λ_{ac}^{\{1\}}+ λ_{bd}^{\{2\}}) θ(γ)}{θ(λ_{ac}^{\{1\}}) θ(λ_{bd}^{\{2\}})} t̂_{cb} t̂_{ad} ,

(A.6)  \frac{θ(λ_{bd}^{\{2\}}+ γ) θ(λ_{bd}^{\{2\}}− γ)}{(θ(λ_{bd}^{\{2\}}))^{2}} t̂_{ab} t̂_{cd} − \frac{θ(λ_{ac}^{\{1\}}+ γ) θ(λ_{ac}^{\{1\}}− γ)}{(θ(λ_{ac}^{\{1\}}))^{2}} t̂_{cd} t̂_{ab} = \frac{θ(λ_{ac}^{\{1\}}+ λ_{bd}^{\{2\}}) θ(γ)}{θ(λ_{ac}^{\{1\}}) θ(λ_{bd}^{\{2\}})} t̂_{ad} t̂_{cb} ,

t̂_{ad} t̂_{cb} − \frac{θ(λ_{ac}^{\{1\}}+ γ) θ(λ_{ac}^{\{1\}}− γ)}{(θ(λ_{ac}^{\{1\}}))^{2}} t̂_{cb} t̂_{ad} = − \frac{θ(λ_{ac}^{\{1\}}− λ_{bd}^{\{2\}}) θ(γ)}{θ(λ_{ac}^{\{1\}}) θ(λ_{bd}^{\{2\}})} t̂_{ab} t̂_{cd} , (A.7)

t̂_{ad} t̂_{cb} − \frac{θ(λ_{bd}^{\{2\}}+ γ) θ(λ_{bd}^{\{2\}}− γ)}{(θ(λ_{bd}^{\{2\}}))^{2}} t̂_{cb} t̂_{ad} = − \frac{θ(λ_{ac}^{\{1\}}− λ_{bd}^{\{2\}}) θ(γ)}{θ(λ_{ac}^{\{1\}}) θ(λ_{bd}^{\{2\}})} t̂_{cd} t̂_{ab} .

All these formulae follow from (3.2) – (3.5) and (A.1), and the summation formulae for the theta function. Combining formula (A.5) with formulae (A.3) and (A.4) for k = 1 we can obtain the following Serre-type relations:

(A.8)  θ(λ_{ac}^{{1}}+ λ_{bd}^{{2}}) θ(λ_{ac}^{{1}}− γ) θ(λ_{bd}^{{2}}− 2γ) θ(γ) t̂_{ab}^{2} t̂_{cd} −
− θ(λ_{ac}^{{1}}+ λ_{bd}^{{2}}− γ) θ(λ_{ac}^{{1}}) θ(λ_{bd}^{{2}}− γ) θ(2γ) t̂_{ab} t̂_{cd} t̂_{ab} +
+ θ(λ_{ac}^{{1}}+ λ_{bd}^{{2}}− 2γ) θ(λ_{ac}^{{1}}+ γ) θ(λ_{bd}^{{2}}) θ(γ) t̂_{cd} t̂_{ab}^{2} = 0 ,

(A.9)  θ(λ_{ac}^{{1}}+ λ_{bd}^{{2}}+ 2γ) θ(λ_{ac}^{{1}}) θ(λ_{bd}^{{2}}− γ) θ(γ) t̂_{ab} t̂_{cd}^{2} −
− θ(λ_{ac}^{{1}}+ λ_{bd}^{{2}}+ γ) θ(λ_{ac}^{{1}}+ γ) θ(λ_{bd}^{{2}}) θ(2γ) t̂_{cd} t̂_{ab} t̂_{cd} +
+ θ(λ_{ac}^{{1}}+ λ_{bd}^{{2}}) θ(λ_{ac}^{{1}}+ 2γ) θ(λ_{bd}^{{2}}+ γ) θ(γ) t̂_{cd}^{2} t̂_{ab} = 0 .

Lemma A.1. Let a < c and b < d. Then the following relations hold:

t̂_{ab} t̂_{cd}^{k} − t̂_{cd}^{k} t̂_{ab} = \frac{θ(λ_{ac}^{\{1\}}+ λ_{bd}^{\{2\}}+ (k − 1)γ) θ(kγ)}{θ(λ_{ac}^{\{1\}}) θ(λ_{bd}^{\{2\}}+ (k − 1)γ)} t̂_{cb} t̂_{cd}^{k−1}t̂_{ad} ,

(A.10)  t̂_{ab}^{k} t̂_{cd} − t̂_{cd} t̂_{ab}^{k} = \frac{θ(λ_{ac}^{\{1\}}+ λ_{bd}^{\{2\}}− (k − 1)γ) θ(kγ)}{θ(λ_{ac}^{\{1\}}− (k − 1)γ) θ(λ_{bd}^{\{2\}})} t̂_{cb} t̂_{ab}^{k−1}t̂_{ad} ,

t̂_{ad} t̂_{cb}^{k} − \frac{θ(λ_{ac}^{\{1\}}+ kγ) θ(λ_{ac}^{\{1\}}− γ) θ(λ_{bd}^{\{2\}}− (k − 1)γ)}{(θ(λ_{ac}^{\{1\}}))^{2} θ(λ_{bd}^{\{2\}})} t̂_{cb}^{k} t̂_{ad} = (A.11)

= − \frac{θ(λ_{ac}^{\{1\}}− λ_{bd}^{\{2\}}+ (k − 1)γ) θ(kγ)}{θ(λ_{ac}^{\{1\}}) θ(λ_{bd}^{\{2\}})} t̂_{cb}^{k−1}t̂_{ab} t̂_{cd} .

Proof. For k = 1 these formulae coincide with formulae (A.5) and (A.7), respectively. All the proofs for k > 1 are similar to each other. So we will prove only formula (A.11).

Multiply formula (A.6) by t̂_{cb}^{k−1} from the right, and push the factor t̂_{cb}^{k−1} in the left hand side through all the products from right to left using relations (A.3). Taking into account formula (A.5) we get

t̂_{ad} t̂_{cb}^{k} − \frac{θ(λ_{ac}^{\{1\}}+ kγ) θ(λ_{ac}^{\{1\}}− γ) θ(λ_{bd}^{\{2\}}− (k − 1)γ)}{(θ(λ_{ac}^{\{1\}}))^{2} θ(λ_{bd}^{\{2\}})} t̂_{cb}^{k} t̂_{ad} =

= (F (λ_{ac}^{{1}}, λ_{bd}^{{2}}) + F (−λ_{bd}^{{2}}, −λ_{ac}^{{1}})) t̂_{cb}^{k−1} t̂_{ab} t̂_{cd}

where

F (u, v) = \frac{θ(u + (k − 1)γ) θ(v + γ) θ(v − kγ)}{θ(u + v) θ(v) θ(γ)} .

F (u, v) + F (−v, −u) is a quasiperiodic function of u with periods 1 and τ and has only simple poles. So, it is completely determined by its multipliers and residues. Hence,

F (u, v) + F (−v, −u) = − \frac{θ(u − v + (k − 1)γ) θ(kγ)}{θ(u) θ(v)}

□

Appendix B. Quantum determinant

The construction of Det T (u) and the proof of Proposition 2.1 can be obtained by extending the standard fusion procedure technique to the dynamical case.

For any k = 2, …, N let A_{k} be the complete antisymmetrizer in (ℂ^{N})^{⊗k}:

A_{k}(x_{1} ⊗ … ⊗ x_{k}) = \frac{1}{k!} ∑_{i∈S_{k}}sign(i) (x_{i_{1}} ⊗ … ⊗ x_{i_{k}}) .

Set A = A_{2} and let S = 1 − A be the corresponding symmetrizer. The R-matrix R(u, λ) has a simple pole at u = γ. Denote by Q(λ) the resique of R(u, λ) at u = γ.

Lemma B.1. Ker Q(λ) = Ker A = Im S.

Due to the inversion relation (1.5) we can write relation (2.3) in the following form:

(B.1)  T^{(13)}(u)T^{(23)}(v)R^{(21)}(u − v, λ) = R^{(21)}(u − v, λ − γh^{(3)})T^{(23)}(v)T^{(13)}(u) .

Then by Lemma B.1 we have that A^{(12)} T^{(23)}(u + γ)T^{(13)}(u) S^{(12)} = 0 , which is equivalent to each of the following relations:

(B.2)  T^{(23)}(u + γ)T^{(13)}(u) S^{(12)} = S^{(12)} T^{(23)}(u + γ)T^{(13)}(u) S^{(12)} ,
A^{(12)} T^{(23)}(u + γ)T^{(13)}(u) = A^{(12)} T^{(23)}(u + γ)T^{(13)}(u) A^{(12)} .

Formula (B.2) shows that for any i = 1, …, k − 1 we have

T^{(k,k+1)}(u + (k − 1)γ) … T^{(1,k+1)}(u) S^{(i,i+1)} =
= S^{(i,i+1)} T^{(k,k+1)}(u + (k − 1)γ) … T^{(1,k+1)}(u) S^{(i,i+1)} ,

which implies that

(B.3)  A_{k}^{(1,...,k)} T^{(k,k+1)}(u + (k − 1)γ) … T^{(1,k+1)}(u) =
= A_{k}^{(1,...,k)} T^{(k,k+1)}(u + (k − 1)γ) … T^{(1,k+1)}(u) A_{k}^{(1,...,k)}

because Ker A_{k} = ∑_{i=1}^{k−1} Im S^{(i,i+1)} and (ℂ^{N})^{⊗k} = Im A_{k} ⊕ Ker A_{k}. We denote the restriction of

A_{k}^{(1,...,k)} T^{(k,k+1)}(u + (k − 1)γ) … T^{(1,k+1)}(u)

to Im A_{k} ⊗ Fun(V) by T^{∧k}(u) and call it the k-th exterior power of T (u) . Since Im A_{N} is onedimensional, the top exterior power T^{∧N}(u) can be considered as an element of End(Fun(V)).

Lemma B.2. For any permutation j ∈ S_{N} we have

(B.4)  T^{∧N}(u) = sign(j) ∑_{i∈S_{N}}sign(i) T_{i_{N},j_{N}} (u + (N − 1)γ) … T_{i_{2},j_{2}}(u + γ) T_{i_{1},j_{1}}(u) .

Proof. Let v_{1}, …, v_{N} be the standard basis of ℂ^{N}. Then for any j ∈ S_{N} we have

(B.5)  A_{N} (v_{j_{1}} ⊗ … ⊗ v_{j_{N}}) = sign(j) A_{N} (v_{1} ⊗ … ⊗ v_{N}) .

Let v ∈ Fun(V) . By the definition of T^{∧N}(u) and using relation (B.3) we get

A_{N} (v_{j_{1}} ⊗ … ⊗ v_{j_{N}}) ⊗ T^{∧N}(u) v =
= ∑_{i∈S_{N}}A_{N} (v_{i_{1}} ⊗ … ⊗ v_{i_{N}}) ⊗ T_{i_{N},j_{N}} (u + (N − 1)γ) … T_{i_{2},j_{2}}(u + γ) T_{i_{1},j_{1}}(u) v .

By formula (B.5) the expression in the right hand side equals

sign(j)A_{N} (v_{j_{1}} ⊗ … ⊗ v_{j_{N}}) ⊗ ∑_{i∈S_{N}}sign(i) T_{i_{N},j_{N}} (u + (N − 1)γ) … T_{i_{2},j_{2}}(u + γ) T_{i_{1},j_{1}}(u) v ,

which proves the lemma. □

Like in the ordinary linear algebra there is a connection between the complementary exterior powers T^{∧k}(u) and T^{∧(N−k)}(u) and the top exterior power T^{∧N}(u) , cf. Theorem B.4. For any two sequences a = (a_{1}, …, a_{k}) and b = (b_{1}, …, b_{k}) set

T_{ab}^{∧k}(u) = ∑_{i∈S_{k}}sign(i) T_{a_{i_{k}}b_{k}}(u + (k − 1)γ) … T_{a_{i_{2}}b_{2}}(u + γ) T_{a_{i_{1}}b_{1}}(u) .

Lemma B.3. Let i, j ∈ S_{k}, and let a′ = (a_{i_{1}} , …, a_{i_{k}}) , b′ = (a_{j_{1}} , …, a_{j_{k}}) . Then

∧k T_{a′b′}(u) = sign(i) sign(j) T_{ab}^{∧k}(u) .

The proof is similar to the proof of Lemma B.4.

Denote by Y_{k} the set of increasing k-tuples of integers from {1, …, N }. For any a ∈ Y_{k} define a ∈ Y_{N−k} to be the complement of a, that is, {a_{1}, …, a_{k}, ā_{1}, …, ā_{N−k}} = {1, …, N }. Denote by aa the permutation (a_{1}, …, a_{k}, ā_{1}, …, ā_{N−k}) ∈ S_{N} .

Theorem B.4.

(B.6)  sign(aa) ∑_{c∈Y_{k}}sign(cc) T_{ca}^{∧(N−k)}(u − (k − 1)γ) T_{cb}^{∧k}(u) = δ_{ab} T^{∧N}(u) .

The proof is similar to the proof of the analogous formlula in the ordinary linear algebra.

It is clear from relation (2.1) and formula (B.4) that the difference operator T^{∧N}(u) commutes with multiplication by scalar functions. So, there exists a function L^{∧N}(u, λ) ∈ Fun(End(V)) such that

(T^{∧N}(u) v)(λ) = L^{∧N}(u, λ) v(λ)

for any v ∈ Fun(V) . Denote by R^{∧N}(u, λ) the function L^{∧N}(u, λ) for the vector representation of E_{τ,γ}(𝔰𝔩_{N}) with the evaluation point x = 0 .

Lemma B.5.

R^{∧N}(u, λ) = \frac{θ(u + (N − 1)γ)}{θ(u)} ∑_{a=1}^{N} ∏_{{b=1 \atop b≠a}}^{N} \frac{θ(λ_{ab} − γ)}{θ(λ_{ab})} E_{aa} .

Proof. Recall that in the vector representation

(T_{aa}(u) v)(λ) = E_{aa}v(λ − γ ε_{a}) + ∑_{{a,b=1 \atop a≠b}}^{N}α(u, λ_{ab}) E_{bb} v(λ − γ ε_{b})

and (T_{ab}(u) v)(λ) = β(u, λ_{ab}) E_{ba}v(λ − γ ε_{b}) , where α(u, ξ) and β(u, ξ) are given by (1.2). By Lemma (B.2) this implies that R^{∧N}(u, λ) is a linear combination of the matrices E_{11}, …, E_{NN} with some functional coefficients. Moreover, taking formula (B.4) for the permutation j such that j_{1} = a we observe that only the term in the sum with i = j contribute to the coefficient of E_{aa}, which, therefore, can be found easily. □

Proof of Proposition 2.1. From relations (2.3) and (2.1) following the definition of T^{∧N}(u) we find that

(R^{∧N})^{(1)}(u − v, λ − γh^{(2)}) (T^{∧N})^{(2)}(u) T^{(12)}(v) = T^{(12)}(v) (T^{∧N})^{(2)}(u) (R^{∧N})^{(1)}(u − v, λ + γh^{(1)}) .

By Lemma B.5 this is equivalent to

∏_{{c=1 \atop c≠a}}^{N} \frac{θ(λ_{ac} − γh_{ac} − γ)}{θ(λ_{ac} − γh_{ac})} T^{∧N}(u) T_{ab}(v) = ∏_{{c=1 \atop c≠b}}^{N} \frac{θ(λ_{bc} − γ)}{θ(λ_{bc})} T_{ab}(v) T^{∧N}(u) .

for any a, b = 1, …, N . Since Det T (u) = \frac{Θ(λ)}{Θ(λ − γh)} T^{∧N}(u) where Θ(λ) = ∏_{1⩽a<b⩽N}θ(λ_{ab}) , the

proposition is proved. □

Appendix C. Multiplicative forms

In this section we essentially follow [EV1, Section 1.4]. Notice that all over the paper the words cocycle and coboundary mean 1-cocycle and 1-coboundary, respectively.

Let I_{k} be the set of k-tuples of pairwise distinct integers from {1, …, N }. A multiplicative k-form f̄ is a map I_{k} → Fun(ℂ) , f̄: a ↦ f_{a}, such that for any a ∈ I_{k} and any i = 1, …, k we have

f_{a_{1},..., a_{k}}(λ) f_{a_{1},..., a_{i−1}, a_{i+1}, a_{i}, a_{i+2},..., a_{k}}(λ) = 1

Let Ω^{k} be the set of all multiplicative k-forms. If f̄ and ḡ are multiplicative k-forms, then f̄ḡ : a ↦ f_{a} g_{a} and f̄/ḡ : a ↦ f_{a}/ḡ_{a} are multiplicative k-forms, which defines an abelian group structure on Ω^{k}. The neutral element is the form ¯1 : 1_{a}(λ) = 1 for any a ∈ I_{k}.

For any nonzero function f(λ) and any a = 1, …, N set (δ_{a}f)(λ) = f(λ − γ ε_{a})/f(λ) , and for any f̄ ∈ Ω^{k} define a multiplicative form df̄ ∈ Ω^{k+1} by

(df̄)_{a_{1},..., a_{k+1}}(λ) = ∏_{i=1}^{k+1}((δ_{a_{i}}f_{a_{1},..., a_{i−1}, a_{i+1},..., a_{k+1}})(λ))^{(−1)^{i−1}}.

For any multiplicative form f̄ we have d^{2}f̄ = ¯1 . The multiplicative form f̄ is called a multiplicative cocycle if df̄ = ¯1 , and a multiplicative coboundary if f̄ = dḡ for a suitable multiplicative form ḡ.

For any meromorphic function f(ξ) in one variable the multiplicative 1-form F = (F_{1}, …, F_{N}) :

F_{a}(λ) = ∏_{1⩽c<a}f(λ_{ca}) ∏_{a<c⩽N}(f(λ_{ac} − γ))^{−1},

is a multiplicative 1-cocycle. If f(ξ) = g(ξ + γ)/g(ξ) , then F is a multiplicative 1-coboundary:

F = dG, where G(λ) = ∏_{1⩽b<c⩽N} g(λ_{bc}) .

If (F_{1}, …, F_{N}) is a multiplicative 1-cocycle, then so does (G_{1}, …, G_{N}) : G_{a}(λ) = F_{a}(−λ + γ ε_{a}) .

Appendix D. Proof of Theorem 4.1

In this section we introduce another ordering on generators of eτ,γ^{O}(𝔰𝔩_{N}) , called ordering by rows, and prove the analogue of Theorem 4.1 for the monomials ordered by rows, cf. Theorem D.1. Since the number or ordered by rows monomials of degree k equals the number of normally ordered monomials of the same degree, and each monomial can be transformed to a linear combination of normally ordered monomials, Theorem D.1 implies Theorem 4.1. We introduce ordered by rows monomials for technical reason because this allows us to reduce the number of cases to be examined at some stage of the proof.

Consider the ordering by rows of generators: t_{ab} ≺ t_{cd} if a < c, or a = c and b < d. Say that the monomial t_{a_{1}b_{1}} … t_{a_{k}b_{k}} is ordered by rows if t_{a_{i}b_{i}} ≺ t_{a_{j}b_{j}} for any i < j, or k = 0 .

Theorem D.1. For any k ∈ ℤ_{⩾0} the ordered by rows monomials of degree k form a basis of 𝔢_{k} over Fun^{⊗2}(ℂ) .

Proof. To save words from now on we write ordered instead of ordered by rows. We call an equality

disordered monomial = linear combination of ordered monomials

an ordering rule for the monomial in the left hand side. The commutation relations (3.3) – (3.5) have the important property: A. Any relation is a linear combinations of ordering rules, and the complete list of linear independent

ordering rules contains precisely one rule for each disordered product of generators.

For k = 0 and k = 1 the claim of Theorem D.1 is immediate. Let k > 1 . First we prove that the ordered monomials of degree k span 𝔢_{k} over Fun^{⊗2}(ℂ) . Indeed, one can transform any monomial t_{a_{1}b_{1}} … t_{a_{k}b_{k}} to a linear combination of ordered monomials by the following procedure. Pick up any disordered product of adjacent factors and replace it by a sum of ordered products using the ordering rule, then do the same for each of the obtained monomials. To see that the procedure terminates and, hence, produces a linear combination of ordered monomials, introduce auxilary gradings on monomials by the rule

r(t_{a_{1}b_{1}} … t_{a_{k}b_{k}}) = ∑_{i=1}^{k} ia_{i} , r′(t_{a_{1}b_{1}} … t_{a_{k}b_{k}}) = ∑_{i=1}^{k} ib_{i} ,

and observe that at each nontrivial step of the procedure we replace a monomial by a sum of monomials of either less degree r, or the same degree r and less degree r′. We call the described procedure a regular transformation of the monomial t_{a_{1}b_{1}} … t_{a_{k}b_{k}} to a linear combination of ordered monomials.

If an ordering rule is applied to a product t_{a_{i}b_{i}}t_{a_{i+1}b_{i+1}} in the monomial t_{a_{1}b_{1}} … t_{a_{k}b_{k}} , then we say that the ordering rule is applied at the i-th place.

By the standard reasoning the property A of the commutation relations (3.3) – (3.5) implies that Theorem D.1 follows from Proposition D.2. □

Proposition D.2. Any regular transformation of the monomial t_{a_{1}b_{1}} … t_{a_{k}b_{k}} produces the same linear combination of the ordered monomials.

Proof. For k = 2 the claim follows from the property A. For k = 3 the claim can be verified in a straightforward way. We discuss more details of k = 3 case at the end of the proof.

Let k > 3 . For the proof we use induction with respect to the lexicographical ordering on monomials defined by pair of degrees (r, r′) . The claim of the proposition is obvious for the ordered monomials, which provides the base of induction.

Let I and II be two regular transformations of the monomial t_{a_{1}b_{1}} … t_{a_{k}b_{k}} to linear combinations of ordered monomials. If for I and II the first steps coincide, then they produce the same results by the induction assumption. Otherwise, let us construct two additional regular transformations III and IV such that the first steps of I and III coincide, the same holds for II and IV , and III and IV produce the same results. By the previous remark, this proves the proposition.

Assume that for the transformation I an ordering rule at the first step is applied at the i-th place, and for the transformation II at the j-th place. If |i − j| > 1 then we define the transformation III as follows: first apply an ordering rule at the i-th place, next apply an ordering rule at the j-th place for all monomials obtained at the first step, then continue in any possible way. The transformation IV is defined similarly with i and j interchanged. By the induction assumption the transformations III and IV produce the same results because after the first two steps of both III and IV one has identical linear combinations of monomials, each of them being lexicographically smaller than the initial monomial t_{a_{1}b_{1}} … t_{a_{k}b_{k}} .

The cases j = i ± 1 are similar. Assume for example that j = i + 1 . Define the transformation III as follows: apply a regular transformation of the product t_{a_{i}b_{i}} t_{a_{i+1}b_{i+1}} t_{a_{i+2}b_{i+2}} to a linear combination of ordered triple products, making the first step at the i-th place, then continue in any possible way. Define the transformation IV similarly, but making the first step at the (i + 1)-th place. Then the claim of the proposition for k = 3 shows that after the first stages of both III and IV one has identical linear combinations of monomials, each of them being lexicographically smaller than the initial monomial t_{a_{1}b_{1}} … t_{a_{k}b_{k}} . Therefore, by the induction assumption the transformations III and IV produce the same results.

It remains to complete the proof for k = 3 . For any monomial t_{a_{1}b_{1}} t_{a_{2}b_{2}} t_{a_{3}b_{3}} there are at most two regular transformations, and the regular transformation is unique unless t_{a_{3}b_{3}} ≺ t_{a_{2}b_{2}} ≺ t_{a_{1}b_{1}} . The rest of the proof is given by the straightforward calculations. The simplest cases appear when a_{1} = a_{2} = a_{3} or b_{1} = b_{2} = b_{3} . We present below the most bulky example when a_{1} > a_{2} > a_{3} and b_{1}, b_{2}, b_{3} are pairwise distinct. □

Example. Regular transformations of the monomial t_{34} t_{25} t_{16} . Let functions α(u, ξ) and β(u, ξ) be given by (1.2). Making the first step at the first place we have

t_{34} t_{25} t_{16} = α(λ_{32}^{{1}}, λ_{54}^{{2}}) t_{25} t_{34} t_{16} − β(λ_{32}^{{1}}, λ_{45}^{{2}}) t_{24} t_{35} t_{16} =
= α(λ_{32}^{{1}}, λ_{54}^{{2}})(α(λ_{31}^{{1}}, λ_{64}^{{2}}) t_{25} t_{16} t_{34} − β(λ_{31}^{{1}}, λ_{46}^{{2}}) t_{25} t_{14} t_{36}) +
− β(λ_{32}^{{1}}, λ_{45}^{{2}})(α(λ_{31}^{{1}}, λ_{65}^{{2}}) t_{24} t_{16} t_{35} − β(λ_{31}^{{1}}, λ_{56}^{{2}}) t_{24} t_{15} t_{36}) =

= α(λ_{32}^{{1}}, λ_{54}^{{2}}) α(λ_{31}^{{1}}, λ_{64}^{{2}})(α(λ_{21}^{{1}}, λ_{65}^{{2}}) t_{16} t_{25} t_{34} − β(λ_{21}^{{1}}, λ_{56}^{{2}}) t_{15} t_{26} t_{34}) +
− α(λ_{32}^{{1}}, λ_{54}^{{2}}) β(λ_{31}^{{1}}, λ_{46}^{{2}})(α(λ_{21}^{{1}}, λ_{45}^{{2}}) t_{14} t_{25} t_{36} − β(λ_{21}^{{1}}, λ_{54}^{{2}}) t_{15} t_{24} t_{36}) +
− β(λ_{32}^{{1}}, λ_{45}^{{2}}) α(λ_{31}^{{1}}, λ_{65}^{{2}})(α(λ_{21}^{{1}}, λ_{64}^{{2}}) t_{16} t_{24} t_{35} − β(λ_{21}^{{1}}, λ_{46}^{{2}}) t_{14} t_{26} t_{35}) +
+ β(λ_{32}^{{1}}, λ_{45}^{{2}}) β(λ_{31}^{{1}}, λ_{56}^{{2}})(α(λ_{21}^{{1}}, λ_{54}^{{2}}) t_{15} t_{24} t_{36} − β(λ_{21}^{{1}}, λ_{45}^{{2}}) t_{14} t_{25} t_{36}) ,

while making the first step at the second place we have

t_{34} t_{25} t_{16} = α(λ_{21}^{{1}}, λ_{65}^{{2}}) t_{34} t_{16} t_{25} − β(λ_{21}^{{1}}, λ_{56}^{{2}}) t_{34} t_{15} t_{26} =
= α(λ_{21}^{{1}}, λ_{65}^{{2}})(α(λ_{31}^{{1}}, λ_{64}^{{2}}) t_{16} t_{34} t_{25} − β(λ_{31}^{{1}}, λ_{46}^{{2}}) t_{14} t_{36} t_{25}) +
− β(λ_{21}^{{1}}, λ_{56}^{{2}})(α(λ_{31}^{{1}}, λ_{54}^{{2}}) t_{15} t_{34} t_{26} − β(λ_{31}^{{1}}, λ_{45}^{{2}}) t_{14} t_{35} t_{26}) =
= α(λ_{21}^{{1}}, λ_{65}^{{2}}) α(λ_{31}^{{1}}, λ_{64}^{{2}})(α(λ_{32}^{{1}}, λ_{54}^{{2}}) t_{16} t_{25} t_{34} − β(λ_{32}^{{1}}, λ_{45}^{{2}}) t_{16} t_{24} t_{35}) +
− α(λ_{21}^{{1}}, λ_{65}^{{2}}) β(λ_{31}^{{1}}, λ_{46}^{{2}})(α(λ_{32}^{{1}}, λ_{56}^{{2}}) t_{14} t_{25} t_{36} − β(λ_{32}^{{1}}, λ_{65}^{{2}}) t_{14} t_{26} t_{35}) +
− β(λ_{21}^{{1}}, λ_{56}^{{2}}) α(λ_{31}^{{1}}, λ_{54}^{{2}})(α(λ_{32}^{{1}}, λ_{64}^{{2}}) t_{15} t_{26} t_{34} − β(λ_{32}^{{1}}, λ_{46}^{{2}}) t_{15} t_{24} t_{36}) +
+ β(λ_{21}^{{1}}, λ_{56}^{{2}}) β(λ_{31}^{{1}}, λ_{45}^{{2}})(α(λ_{32}^{{1}}, λ_{65}^{{2}}) t_{14} t_{26} t_{35} − β(λ_{32}^{{1}}, λ_{56}^{{2}}) t_{14} t_{25} t_{36}) .

The coefficients of the monomials t_{16} t_{25} t_{34} , t_{15} t_{26} t_{34} and t_{16} t_{24} t_{35} coincide identically. To check that the coefficients for other monomials are the same one should take into account the explicit form of α(u, ξ) and β(u, ξ) and use summation formulae for the theta function.

Appendix E. Elliptic quantum group e_{τ,γ}(𝔰𝔩_{2})

This section is an illustration of general constructions in the simplest case N = 2 . In this case ω_{1} = ε_{1} = −ε_{2} = ρ = α_{1}/2 and 𝔥^{∗} = ℂω_{1} . For any λ ∈ 𝔥^{∗} we have λ_{1} = −λ_{2} = λ_{12}/2 and λ = λ_{12} ω_{1} . We identify functions on 𝔥^{∗} with functions of one variable λ_{12} .

The operator algebra eτ,γ^{O}(𝔰𝔩_{2}) is generated over ℂ by elements t_{11}, t_{12}, t_{21}, t_{22} and functions f ∈ Fun^{⊗2}(ℂ) . According to Theorem 4.1 monomials t_{21}^{k_{21}} t_{11}^{k_{11}} t_{22}^{k_{22}} t_{12}^{k_{12}} , k_{11}, k_{12}, k_{21}, k_{22} ∈ ℤ_{⩾0} , form a basis of eτ,γ^{O}(𝔰𝔩_{2}) as a Fun^{⊗2}(ℂ)-module.

Let Q̂ = (Q_{1}, Q_{2}) be a multiplicative cocycle, which means in this case that

Q_{1}(λ_{12}) Q_{2}(λ_{12} − γ) = Q_{1}(λ_{12} + γ) Q_{2}(λ_{12}) .

A Verma module M_{μ,Q̂} of highest weight μ ∈ 𝔥^{∗} and dynamical highest weight Q̂ over e_{τ,γ}(𝔰𝔩_{2}) is

constructed as follows. As an 𝔥-module M_{μ,Q̂} = ⨁_{k=0}^{∞} ℂv_{μ,Q̂}[k] , the vector v_{μ,Q̂}[k] having weight

μ − kα_{1} . Notice that v_{μ,Q̂} = v_{μ,Q̂}[0] . The generators t̂_{11}, t̂_{12}, t̂_{21}, t̂_{22} act in Fun(M_{μ,Q̂}) by the rule

t̂_{21} v_{μ,Q̂}[k] = v_{μ,Q̂}[k + 1] ,

t̂_{11} v_{μ,Q̂}[k] = Q_{1}(λ_{12} + kγ) \frac{θ(λ_{12} + kγ)}{θ(λ_{12})} v_{μ,Q̂}[k] ,

t̂_{22} v_{μ,Q̂}[k] = Q_{2}(λ_{12} + kγ) \frac{θ(λ_{12} − (μ_{12} − k)γ)}{θ(λ_{12} − (μ_{12} − 2k)γ)} v_{μ,Q̂}[k] ,

t̂_{12} v_{μ,Q̂}[k] = Q_{1}(λ_{12} + kγ) Q_{2}(λ_{12} + (k − 1)γ) \frac{θ((μ_{12} − k + 1)γ) θ(kγ)}{θ(λ_{12}) θ(λ_{12} − (μ_{12} − 2k + 2)γ)} v_{μ,Q̂}[k − 1] ,

see (8.1), (8.2) and Lemma 8.1. Taking into account relations (4.1) and (3.6), and Corollary 3.4, one reproduces from these formulae the construction, given in [FV1], of the evaluation Verma module over E_{τ,γ}(𝔰𝔩_{2}) tensored with a one-dimensional representation of E_{τ,γ}(𝔰𝔩_{2}) .

In general, to compute the dynamical Shapovalov form on eτ,γ^{O}(𝔰𝔩_{2}) one has to find

(E.1)  S(f(λ_{12}^{{1}}, λ_{12}^{{2}})) t̂_{21}^{k_{21}} t̂_{11}^{k_{11}} t̂_{22}^{k_{22}} t̂_{12}^{k_{12}}, g(λ_{12}^{{1}}, λ_{12}^{{2}}) t̂_{21}^{m_{21}} t̂_{11}^{m_{11}} t̂_{22}^{m_{22}} t̂_{12}^{m_{12}})

for any f , g ∈ Fun^{⊗2}(ℂ) and any monomials t̂_{21}^{k_{21}} t̂_{11}^{k_{11}} t̂_{22}^{k_{22}} t̂_{12}^{k_{12}} , t̂_{21}^{m_{21}} t̂_{11}^{m_{11}} t̂_{22}^{m_{22}} t̂_{12}^{m_{12}} . The answer is zero unless k_{12} = m_{12} = 0 and k_{21} = m_{21} . Moreover, contributions of the functions f , g and factors t_{aa} are easy to calculate, and we find that in the nonzero case expression (E.1) equals

f(−λ_{12}^{{2}} + (k_{11} − k_{21} − k_{22})γ , −λ_{12}^{{1}} + (k_{11} + k_{21} − k_{22})γ) ×
× g(λ_{12}^{{1}} − (k_{11} + k_{21} − k_{22})γ, λ_{12}^{{2}} − (k_{11} − k_{21} − k_{22})γ) q_{1}^{k_{11}} q_{2}^{k_{22}} S(t̂_{21}^{k_{21}} , t̂_{21}^{k_{21}}) q_{1}^{m_{11}} q_{2}^{m_{22}}

where

S(t̂_{21}^{k} , t̂_{21}^{k}) = (−1)^{k} ∏_{j=0}^{k−1} \frac{θ(λ_{12}^{\{1\}}− λ_{12}^{\{2\}}− jγ)) θ((j + 1)γ)}{θ(λ_{12}^{\{1\}}− jγ) θ(λ_{12}^{\{2\}}+ jγ)} q_{1}^{k} q_{2}^{k} , k ∈ ℤ_{⩾0} .

For the corresponding part of the dynamical Shapovalov pairing for M_{μ,Q̂} we have

S_{μ,Q̂}(t̂_{21}^{k} , v_{μ,Q̂}[k]) = (−1)^{k} ∏_{j=0}^{k−1} (Q_{1}(λ_{12} + (j + 1)γ) Q_{2}(λ_{12} + jγ) \frac{θ((μ_{12} − j)γ) θ((j + 1)γ)}{θ(λ_{12} − jγ) θ(λ_{12} − (μ_{12} − j)γ)} ) .

The contravariant form C_{μ,Q̂} : Fun(M_{μ,Q̂}) ⊗_{ℂ} Fun(M_{μ,Q}) → Fun(ℂ) is given by

C_{μ,Q̂}(v_{μ,Q̂}[k], v_{μ,Q}[l]) = δ_{kl} (−1)^{k} ∏_{j=0}^{k−1} (Q_{1}(λ_{12} + (j + 1)γ) Q_{2}(λ_{12} + jγ) ×

× \frac{θ((μ_{12} − j)γ) θ((j + 1)γ)}{θ(λ_{12} + (j + 1)γ) θ(λ_{12} − (μ_{12} − j − k)γ)} ) .

Let γ ∉ ℚ + τ ℚ and assume that Q̂ is nondegenerate. Then the module M_{μ,Q̂} is irreducible provided that μ_{12} ∉ ℤ_{⩾0} . If μ_{12} ∈ ℤ_{⩾0} , then v_{μ,Q̂}[μ_{12} + 1] is a regular singular vector generating an irreducible submodule N_{μ,Q̂} isomorphic to M_{−μ−2ρ,Q} where Q(λ_{12}) = Q̂(λ_{12} − μ_{12} − 2) . The quotient module V_{μ,Q̂} = M_{μ,Q̂}/N_{μ,Q̂} is the irreducible highest weight e_{τ,γ}(𝔰𝔩_{2})-module with highest weight μ and dynamical highest weight Q̂, and it has dimension μ_{12} + 1 , the same as the irreducible 𝔰𝔩_{2}-module of highest weight μ.

Appendix F. Proof of Theorem 10.5

In [TV2, Section 2.6] for any semisimple Lie algebra 𝔤 we defined rational functions B_{w}(λ) of λ labeled by elements of the Weil group. The functions takes values in a certain completion of U(𝔤) . Here we need the function B_{w}(λ − ρ) for the particular case of 𝔤 = 𝔰𝔩_{N} and w = w_{X} , the longest element of the Weil group. For brevity we denote this function by B_{X}(λ) . We list the required properties of B_{X}(λ) below using the notation of the present paper. All of them easily follow from the properties of B_{w}(λ) given in [TV2].

Consider the following series

G_{ab}(λ) = ∑_{k=0}^{∞} e_{ba}^{k}e_{ab}^{k} ∏_{j=1}^{k} \frac{1}{j (λ_{ab} − e_{aa} + e_{bb} − j)} .

Then B_{X}(λ) equals the ordered product ∏_{1⩽a<b⩽N}G_{ab}(λ) where the factor G_{ab} is to the left from the

factor G_{cd} if a < c, or a = c and b > d. For instance, if N = 3 , then B_{X}(λ) = G_{12}(λ)G_{13}(λ)G_{23}(λ) .

The series B_{X}(λ) acts in any finite-dimensional 𝔰𝔩_{N}-module V , commuting with the 𝔥-action. The action of B_{X}(λ) gives an element of Rat(End(V)), which tends to 1 as λ goes to infinity in a generic direction and, therefore, is invertible for generic λ.

Proposition F.1. For any finite-dimensional 𝔰𝔩_{N}-modules V, W we have

B_{X}(λ)|_{V ⊗W} (X ⊗ X)J_{V W} (w_{X}(λ))(X ⊗ X)^{−1} = J_{V W} (λ)(B_{X}(λ − h^{(2)}) ⊗ B_{X}(λ)) .

Corollary F.2. R_{V W} (λ − ρ) = (B_{X}(λ − h^{(2)}) ⊗ B_{X}(λ))^{−1}R_{V W} (λ)(B_{X}(λ) ⊗ B_{X}(λ − h^{(1)})).

The statement follows from Corollary 9.8 and cocommutativity of the coproduct Δ .

Lemma F.3. Let U be the vector representation of 𝔰𝔩_{N} . Then B_{X}(λ)|_{U} = ∑_{a=1}^{N} E_{aa} ∏_{1⩽b<a}(1 + λ_{ba}^{−1}) .

Proof of Theorem 10.5. By the definition of isomorpic functors the assertion of the theorem means that for any finite-dimensional 𝔰𝔩_{N}-module V there exists an isomorphism ψ_{V} ∈ Mor(Ẽ(V), E(V)), and for any morphism φ : V → W of 𝔰𝔩_{N}-modules one has E(φ) ∘ ψ_{V} = ψ_{W} ∘ Ẽ(φ) .

It follows from formula (10.8), Proposition 10.2, Corollary F.2 and Lemma F.3 that B_{X}(λ)|_{V} is

a morphism from Ẽ(V) to E(V) . It is an isomorphism since B_{X}(λ)|_{V} is invertible for generic λ.

Moreover, for any morphism φ : V → W of 𝔰𝔩_{N}-modules we have φ B_{X}(λ)|_{V} = B_{X}(λ)|_{W} φ. Since

both E and Ẽ send φ to the constant function φ ∈ Rat(Hom(V, W)), the theorem is proved. □

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