Quantized moduli spaces of the bundles on the elliptic curve

and their applications.

by A.V.Odesskii and B.L.Feigin

Introduction

Let 𝔟 be the Borel subalgebra of some Kac-Moody algebra 𝔤 and B the corresponding group. (If 𝔟 is infinite-dimensional, then B is a projective limit of finite-dimensional groups). Let M(E, 𝔟) be the moduli space of B-bundles on the elliptic curve E. There is the natural map B → H, where H is a Cartan subgroup, so each B-bundle defines an H-bundle. Note, that each H-bundle on E has the characteristic class which lies in H^{2}(E, L) ≅ L, where L is the root lattice of Kac-Moody algebra 𝔤. For each n ∈ L we denote by M_{n}(E, 𝔟) the corresponding connected component of M(E, 𝔟).

In [3] we define the Hamiltonian structure on the manifold M(E, 𝔟) and quantize the coordinate ring of the manifold M_{n}(E, 𝔟) in the case when n = ∑_{i} n_{i}δ_{i}, {δ_{i}} are

simple positive roots, {n_{i}} are positive and big enough. Actually, our construction works in case of arbitrary {n_{i}}.

In this paper we consider the case when almost all n_{i} are zero and some of them may be negative. Discuss first the case when all {n_{i}} are zero. If 𝔤 is finitedimensional, then M_{0}(E, 𝔟) is trivial. If 𝔤 is an affine Kac-Moody algebra, then M_{0}(E, 𝔟) is infinite-dimensional with trivial Hamiltonian structure. Let 𝔤 = 𝔞̂, where 𝔞 is a finite-dimensional semisimple Lie algebra and P be a moduli space of 𝔞-bundles on E. Then M_{0}(E, 𝔟) is a space of maps φ : D → P, where D is a formal neighborhood of the origin in ℂ such that φ(0) is a trivial bundle. In §5 we study the construction in this case. In the case 𝔤 = ŝl_{h} we write the quantization of M_{0}(E, 𝔟) explicitly. It means that we construct an infinite set of commuting elements in some algebra.

In §4.2 we consider the case when 𝔤 = sl_{h+1} and the set (n_{1}, …, n_{h}) is (m, 0, 0, …, 0). The moduli space M_{(m,0,...,0)}(E, 𝔟) has a projection on E^{h} and a typical fiber is (ℂP^{m−1})^{h}. After quantization we get a new construction of elliptic R-matrix.

In the §4.4 we consider again the case 𝔤 = sl_{h+1}, when the set of {n_{i}} is (1, 0, …, 0, m). In this case our algebra contains the Sklyanin subalgebra Q_{hm+1,h}(E, τ) (see[2]).

In the §4.5 again for 𝔤 = sl_{h+1} we consider the set (0, …, 0, 1, 0, …, 0). In this case we get the generalized elliptic R-matrix (compare [2], formula (6)).

In the §4.3 we consider the case (−1, 0, …, 0, m) for 𝔤 = sl_{h+1}. It gives us the quantization of the coordinate ring of the Grassmannian of h-dimensional planes in m − h + 1-dimensional space.

Now we describe the contents of the paper.

In §1 we introduce the notations and construct the algebra Q_{n,Δ}(E, τ). Here Δ is a root system of 𝔤, τ ∈ ℂ. For τ = 0 this algebra is the coordinate ring of the manifold M_{n}(E, 𝔟) and for generic τ it is a quantization of this coordinate ring.

In §2 we construct some representations of the algebras Q_{n,Δ}(E, τ) and in §3 we study intertwining operators for these representations. Namely, we embed the algebra Q_{n′,Δ}(E, −τ) for some n′ into the algebra of intertwining operators.

In §4 we apply the construction of the algebra Q_{n,Δ}(E, τ) to produce some objects: the elliptic Belavin R-matrix, the quantization of the algebra of functions on the Grassmannian, the algebra Q_{hm+1,h}(E, τ) (see [2] for definition), some generalized elliptic R-matrix.

In §5 we consider the case of affine 𝔤. We write down the explicit formula for commuting elements in the case 𝔤 = ŝl_{h}.

In §6 we construct the elliptic deformation of the Poison algebra and write down the explicit formula for commuting elements in this case (see also [5]).

§1. Algebra Q_{n,Δ}(E, τ).

1.1. Notations. Let Δ be the root system of the Kac-Moody algebra 𝔤, Δ^{+} are the positive roots, {δ_{1}, …, δ_{h}} are the simple positive roots (so h is rank 𝔤). We suppose that the set of simple positive roots in Δ is fixed. Let (a_{i,j}), 1 ⩽ i, j ⩽ h be the Cartan matrix, a_{i,j} = \frac{2(δ_{i},δ_{j})}{(δ_{i},δ_{i})}. Actually, for our construction of Q_{n,Δ}(E, τ), we need only a linear space with a fixed basis {δ_{1}, …, δ_{h}} and a scalar product (δ_{i}, δ_{j}). But we don’t know anything about the properties of the algebra Q_{n,Δ}(E, τ) if {δ_{1}, …, δ_{h}} is not a set of simple positive roots of the finite dimensional or affine Kac-Moody algebra. Let L be a lattice with the basis {δ_{1}, …, δ_{h}}, L ≅ ℤ^{h} and let L^{+} = {l_{1}δ_{1} + … + l_{h}δ_{h}; l_{1}, …, l_{h} ⩾ 0} ⊂ L.

Let us n : L → ℤ be homomorphism of the abelian groups, so n_{i} = n(δ_{i}) ∈ ℤ. Let E = ℂ⧸Γ be an elliptic curve, where Γ = {m_{1} + m_{2}η; m_{1}, m_{2} ∈ ℤ} is a

lattice, Im η > 0. For m ∈ ℤ, c ∈ ℂ we denote by ξ_{m,c} a linear bundle on E such that its sections are realized as functions on ℂ with the following properties:

f(z + 1) = f(z); f(z + η) = e^{−2πi(mz+c)}f(z)  (1)

It is clear that ξ_{m,c} ≅ ξ_{m′,c′} if and only if m = m′ and c − c′ ∈ Γ. The space of holomorphic sections H^{0}(ξ_{m,c}) is realized as a space of holomorphic functions on ℂ satisfying (1). We denote this space by Θ_{m,c}(Γ). It is easy to see that dim Θ_{m,c}(Γ) = m if m > 0, Θ_{m,c}(Γ) = 0 if m < 0. If m = 0 then Θ_{0,c}(Γ) = 0 for c ∉ Γ. If m = 0 and c ∈ Γ then the bundle ξ_{0,c} is trivial and dim H^{0}(ξ_{0,c}) = 1. For m > 0 the elements of Θ_{m,c}(Γ) are called θ-functions of order m. It is clear that for m > 0 for different c the spaces Θ_{m,c}(Γ) are isomorphic and can be identified by translation of ℂ. It is easy to check that every θ-function of order m has exactly m zeros mod Γ and the sum of these zeros is equal to c + \frac{1}{2}m mod Γ. Let θ(z) = ∑_{α∈ℤ}(−1)^{α}e^{2πi(αz+ \frac{α(α−1)}{2} η)}. It is clear that θ(z) ∈ Θ_{1, \frac{1}{2}} (Γ), θ(0) = 0,

θ(−z) = −e^{−2πiz}θ(z).

1.2. Algebra F_{Δ}(E, τ). Let τ ∈ ℂ. We will construct an associative algebra F_{Δ}(E, τ) that depends on 2 continues parameters: an elliptic curve E and a point τ ∈ ℂ. The algebra F_{Δ}(E, τ) is L^{+}-graded, so F_{Δ}(E, τ) = ⨁_{l∈L^{+}} F_{l} and

F_{l} ∗ F_{l′} ⊂ F_{l+l′}. Here ∗ is the product in the algebra F_{Δ}(E, τ). By definition, the space F_{l} for l = l_{1}δ_{1} + … + l_{h}δ_{h} is a space of everywhere meromorphic functions f(x_{1,1}, …, x_{l_{1},1}; …; x_{1,h}, …, x_{l_{h},h}; u_{1}, …, u_{h}) in l_{1} + … + l_{h} + h variables {x_{α,i}, u_{i}; 1 ⩽ i ⩽ h, 1 ⩽ α ⩽ l_{i}}. We assume, that these functions are symmetric with respect to each group of variables {x_{1,i}, …, x_{l_{i},i}} for all 1 ⩽ i ⩽ h. Let f ∈ F_{l}, g ∈ F_{l′}, where l = l_{1}δ_{1} + … + l_{h}δ_{h}, l′ = l′_{1}δ_{1} + … + l′_{h}δ_{h}. By definition, for the product f ∗ g ∈ F_{l+l′} we have:

f ∗ g(x_{1,1}, …, x_{l_{1}+l′_{1},1}; …; x_{1,h}, …, x_{l_{h}+l′_{h},h}; u_{1}, …, u_{h}) =

\frac{1}{l_{1}!l′_{1}! … l_{h}!l′_{h}!} ∑_{\begin{matrix}σ_{1}∈S_{l_{1}+l′_{1}} \\ ... & ... & ... & ... & ... \\ σ_{h}∈S_{l_{h}+l′_{h}} \end{matrix}} f(x_{σ_{1}(1),1}, …, x_{σ_{1}(l_{1}),1}; …; x_{σ_{h}(1),h}, …, x_{σ_{h}(l_{h}),h}; u_{1}, …, u_{h})×

g(x_{σ_{1}(l_{1}+1),1}, …, x_{σ_{1}(l_{1}+l′_{1}),1}; …; x_{σ_{h}(l_{h}+1),h}, …, x_{σ_{h}(l_{h}+l′_{h}),h}; u_{1}^{(τ)}, …, u_{h}^{(τ)})× (2)

∏_{\begin{matrix}1⩽i,j⩽h, \\ 1⩽α⩽l_{i}, \\ l_{j}+1⩽β⩽l_{j}+l′_{j} \end{matrix}} \frac{θ(x_{σ_{i}(α),i} − x_{σ_{j}(β),j} − (δ_{i}, δ_{j})τ)}{θ(x_{σ_{i}(α),i} − x_{σ_{j}(β),j})}

Here u_{i}^{(τ)} = u_{i} − 2(l, δ_{i})τ for 1 ⩽ i ⩽ h.

Proposition 1. The formula (2) defines in the space F_{Δ}(E, τ) = ⨁_{l∈L^{+}} F_{l} the

structure of the associative algebra.

Proof. By direct calculation.

Remarks. 1. Let τ = 0. In this case the algebra F_{Δ}(E, 0) is commutative and does not depend on E. The formula for f ∗ g takes such a form:

f ∗ g(x_{1,1}, …, x_{l_{1}+l′_{1},1}; …; x_{1,h}, …, x_{l_{h}+l′_{h},h}; u_{1}, …, u_{h}) =

\frac{1}{l_{1}!l′_{1}! … l_{h}!l′_{h}!} ∑_{\begin{matrix}σ_{1}∈S_{l_{1}+l′_{1}} \\ ... & ... & ... & ... & ... \\ σ_{h}∈S_{l_{h}+l′_{h}} \end{matrix}} f(x_{σ_{1}(1),1}, …, x_{σ_{1}(l_{1}),1}; …; x_{σ_{h}(1),h}, …, x_{σ_{h}(l_{h}),h}; u_{1}, …, u_{h})×

g(x_{σ_{1}(l_{1}+1),1}, …, x_{σ_{1}(l_{1}+l′_{1}),1}; …; x_{σ_{h}(l_{h}+1),h}, …, x_{σ_{h}(l_{h}+l′_{h}),h}; u_{1}, …, u_{h}) (3)

Let P_{l} ⊂ F_{l} be a space of functions which does not depend on {u_{1}, …, u_{h}} and are polynomials in variables {x_{α,i}; 1 ⩽ i ⩽ h, 1 ⩽ α ⩽ l_{i}}. It is clear that P =

⨁_{l∈L^{+}} P_{l} is subalgebra in F_{Δ}(E, 0). We have: P_{l} = S^{l_{1}}P_{δ_{1}} ⊗ S^{l_{2}}P_{δ_{2}} ⊗ … ⊗ S^{l_{h}}P_{δ_{h}}, so

P = S^{∗}P_{δ_{1}} ⊗S^{∗}P_{δ_{2}} ⊗…⊗S^{∗}P_{δ_{h}}. Here P_{δ_{i}} is a space of polynomials in one variable. The formula (3) defines the usual product in this tensor product of symmetric algebras.

2. By definition F_{0} is a space of all meromorphic functions in variables {u_{1}, …, u_{h}}. For f, g ∈ F_{0} we have (see (2)) f ∗ g = fg. It is the usual product of functions. So F_{0} is a field. We have F_{0} ∗ F_{l} ⊂ F_{l} and F_{l} ∗ F_{0} ⊂ F_{l} so we have two structures of a F_{0}-vector space on F_{l}. These two structures are connected by the following: for f ∈ F_{0} and g ∈ F_{l} we have: g∗f(u_{1}, …, u_{h}) = f(u_{1}−2(l, δ_{1})τ, …, u_{h}−2(l, δ_{h})τ)∗g (see (2)). For τ = 0, the algebra F_{Δ}(E, 0) is a F_{0}-algebra.

1.3. Algebra Q_{n,Δ}(E, τ). For every linear form n : L → ℤ we will construct a subalgebra Q_{n,Δ}(E, τ) in the algebra F_{Δ}(E, τ). By definition Q_{n,Δ}(E, τ) = ⨁_{l∈L^{+}} Q_{l},

here Q_{l} ⊂ F_{l} is a space of functions satisfied following conditions:

1. f(x_{1,1}, …, x_{l_{h},h}; u_{1}, …, u_{h}) as a function in variables {x_{1,1}, …, x_{l_{h},h}} is holomorphic outside the divisors {x_{α,i} − x_{β,j} = 0; i ≠ j, (δ_{i}, δ_{j}) ≠ 0} and has a pole of order ⩽ 1 on these divisors.

2. For each x_{α,i}, 1 ⩽ i ⩽ h, 1 ⩽ α ⩽ l_{i} f satisfies (1) for m = n_{i}, c = u_{i} − (δ_{i}, l)τ as a function in x_{α,i}. Here n_{i} = n(δ_{i}). So we have:

f(x_{1,1}, …, x_{α,i} + 1, …, x_{l_{h},h}; u_{1}, …, u_{h}) = f(x_{1,1}, …, x_{l_{h},h}; u_{1}, …, u_{h});

f(x_{1,1}, …, x_{α,i} + η, …, x_{l_{h},h}; u_{1}, …, u_{h}) =
e^{−2πi(n_{i}x_{α,i}+u_{i}−(δ_{i},l)τ)}f(x_{1,1}, …, x_{l_{h},h}; u_{1}, …, u_{h})

3. Let 1 ⩽ i, j ⩽ h such that a_{i,j} < 0. Let 1 ⩽ α_{1}, …, α_{−a_{i,j}+1} ⩽ l_{i}, 1 ⩽ β ⩽ l_{j}. Then f(x_{1,1}, …, u_{h}) = 0 on the affine subspace of codimension −a_{i,j} + 1 defined by the following relations:

x_{α_{1},i} − x_{α_{2},i} = (δ_{i}, δ_{i})τ, x_{α_{2},i} − x_{α_{3},i} = (δ_{i}, δ_{i})τ, …, x_{α_{−a_{i,j}} ,i} − x_{α_{−a_{i,j} +1},i} = (δ_{i}, δ_{i})τ,

x_{α_{−a_{i,j} +1},i} − x_{β,j} = (δ_{i}, δ_{j})τ, x_{β,j} − x_{α_{1},i} = (δ_{i}, δ_{j})τ.

4. Let {δ_{i_{1}}, …, δ_{i_{p}}} generates an irreducible component of the root system Δ; μ_{1}, …, μ_{h}; λ_{1}, …, λ_{h}; ν ∈ ℂ and μ_{j} = λ_{j} = 0 if j ∉ {i_{1}, …, i_{p}}, μ_{i_{α}} = −ν, λ_{i_{α}} = n_{i_{α}}ν for 1 ⩽ α ⩽ p. Then f(x_{1,1} +μ_{1}, …, x_{l_{1},1} +μ_{1}; …; x_{1,h} +μ_{h}, …, x_{l_{h},h} + μ_{h}; u_{1} + λ_{1}, …, u_{h} + λ_{h}) = f(x_{1,1}, …, u_{h}).

Proposition 2. The space Q_{n,Δ}(E, τ) is closed with respect to the product ∗ defined by (2). So Q_{n,Δ}(E, τ) is a subalgebra of the algebra F_{Δ}(E, τ).

Proof. By direct calculation it is easy to see that the product ∗ preserves each of the properties 1-4.

Proposition 3. Let Δ_{1} ⊂ Δ is a root subsystem generated by some subset of {δ_{1}, …, δ_{h}}. Then F_{Δ_{1}}(E, τ) ⊂ F_{Δ}(E, τ), Q_{m,Δ_{1}}(E, τ) ⊂ Q_{n,Δ}(E, τ) as graded subalgebras. Here m is a restriction of n to the sublattice generated by Δ_{1}.

Proof. It is evident from the definitions.

Remarks. 1. Q_{0} is a field of meromorphic functions f(u_{1}, …, u_{h}) satisfying the property 4.

2. We have Q_{0} ∗ Q_{l} ⊂ Q_{l} and Q_{l} ∗ Q_{0} ⊂ Q_{l} so Q_{l} has two structures of a Q_{0}- vector space. These two structures are connected in the following way: for f ∈ Q_{0} and g ∈ Q_{l} we have; g ∗ f(u_{1}, …, u_{h}) = f(u_{1} − 2(l, δ_{1})τ, …, u_{h} − 2(l, δ_{h})τ) ∗ g (see (2)), so the dimension of these spaces is the same.

3. From the properties 1 and 2 of the elements from the space Q_{l} it follows that dim_{Q_{0}}Q_{l} is finite for each l ∈ L^{+}.

4. For τ = 0 the algebra Q_{n,Δ}(E, τ) is a commutative Q_{0}-algebra.

5. Q_{δ_{i}} consists of the functions f(x; u_{1}, …, u_{h}) which are holomorphic with respect to x and satisfy the properties 2 and 4. So dim_{Q_{0}}Q_{δ_{i}} = n_{i} if n_{i} > 0 and Q_{δ_{i}} = 0 if n_{i} ⩽ 0.

6. Q_{αδ_{i}} consists of the functions f(x_{1}, …, x_{α}; u_{1}, …, u_{h}) which are holomorphic and symmetric with respect to {x_{1}, …, x_{α}} and satisfy the properties 2 and 4. It is clear that dim_{Q_{0}}Q_{αδ_{i}} = \frac{n_{i}(n_{i}+1)...(n_{i}+α−1)}{α!} if n_{i} > 0 and Q_{αδ_{i}} = 0 if n_{i} ⩽ 0.

7. If (δ_{i}, δ_{j}) = 0 then Q_{αδ_{i}+βδ_{j}} ≅ Q_{αδ_{i}} ⊗_{Q_{0}} Q_{βδ_{j}}.

8. If (δ_{i}, δ_{j}) ≠ 0 (and i ≠ j), than Q_{δ_{i}+δ_{j}} becomes bigger then Q_{δ_{i}} ⊗_{Q_{0}} Q_{δ_{j}} because of the pole part of the functions from Q_{δ_{i}+δ_{j}}. So we have: dim_{Q_{0}}Q_{δ_{i}+δ_{j}} = n_{i}n_{j} + n_{i} + n_{j}, if n_{i}, n_{j} ⩾ 0. If n_{i} or n_{j} < 0, then Q_{δ_{i}+δ_{j}} = 0.

§2. Representations of the algebras Q_{n,Δ}(E, τ).

Let p ∈ L^{+}, p = p_{1}δ_{1} + … + p_{h}δ_{h}. Let A_{p,Δ}(E, τ) be the algebra generated by {e_{α,i}, e_{α,i}^{−1}; 1 ⩽ i ⩽ h, 1 ⩽ α ⩽ p_{i}} and {φ(y_{1,1}, …, y_{p_{1},1}; …; y_{1,h}, …, y_{p_{h},h}; u_{1}, …, u_{h})}, where φ is any meromorphic function in variables {y_{α,i}, u_{i}; 1 ⩽ i ⩽ h, 1 ⩽ α ⩽ p_{i}} such that as a function of {y_{α,i}} it has poles only on the divisors of the form {y_{α,i} − y_{β,j} − (δ_{i}, δ_{j})τ + μ(δ_{i}, δ_{i})τ − ν(δ_{j}, δ_{j})τ = 0; μ, ν ∈ ℤ, (δ_{i}, δ_{j}) ≠ 0}. We assume such the following relations hold:

e_{α,i}e_{β,j} = −e^{2πi(y_{β,j}−y_{α,i})} \frac{θ(y_{α,i} − y_{β,j} − (δ_{i}, δ_{j})τ)}{θ(y_{β,j} − y_{α,i} − (δ_{i}, δ_{j})τ)}e_{β,j}e_{α,i}  (4)

e_{α,i}y_{β,j} = y_{β,j}e_{α,i}, here i ≠ j or α ≠ β;

e_{α,i}e_{α,i}^{−1} = 1, e_{α,i}y_{α,i} = (y_{α,i} + (δ_{i}, δ_{i})τ)e_{α,i}, e_{α,i}u_{j} = (u_{j} − 2(δ_{i}, δ_{j})τ)e_{α,i};
[y_{α,i}, y_{β,j}] = [y_{α,i}, u_{j}] = [u_{i}, u_{j}] = 0.

Let us define the map x : Q_{n,Δ}(E, τ) → A_{p,Δ}(E, τ) by the following. For f ∈ Q_{0}, we put x(f) = f. Recall that Q_{0} ⊂ A_{p,Δ}(E, τ) by definition. For f(x; u_{1}, …, u_{h}) ∈ Q_{δ_{i}} (if n_{i} > 0 ), we put x(f) = ∑_{1⩽α⩽p_{i}} f(y_{α,i}; u_{1}, …, u_{h})e_{α,i}. In general case for

f(x_{1,1}, …, x_{l_{h},h}; u_{1}, …, u_{h}) ∈ Q_{l}, l = l_{1}δ_{1} + … + l_{h}δ_{h}, we define:

x(f) = ∑_{\begin{matrix}φ_{1,1},...,φ_{p_{h},h}⩾0, \\ φ_{1,i}+...+φ_{p_{i},i}=l_{i} for each 1⩽i⩽h \end{matrix}} B_{φ_{1,1},...,φ_{p_{h},h}}e_{1,1}^{φ_{1,1}} … e_{p_{1},1}^{φ_{p_{1},1}} … e_{1,h}^{φ_{1,h}} … e_{p_{h},h}^{φ_{p_{h},h}}

(5)

Here B_{φ_{1,1},...,φ_{p_{h},h}} is a function in variables {y_{α,i}, u_{i}},

B_{φ_{1,1},...,φ_{p_{h},h}} =
f(y_{1,1}, y_{1,1} + (δ_{1}, δ_{1})τ, …, y_{1,1} + (φ_{1,1} − 1)(δ_{1}, δ_{1})τ, …,
y_{p_{h},h}, y_{p_{h},h} + (δ_{h}, δ_{h})τ, …, y_{p_{h},h} + (φ_{p_{h},h} − 1)(δ_{h}, δ_{h})τ; u_{1}, …, u_{h})×
∏_{1⩽i,j⩽h, 1⩽α⩽p_{i},1⩽β⩽p_{j}, 0⩽μ⩽φ_{α,i}−1,0⩽ν⩽φ_{β,j}−1; i⩽j; if i=j, then α⩽β; if i=j,α=β, then μ<ν} \frac{θ(y_{α,i} + μ(δ_{i}, δ_{i})τ − y_{β,j} − ν(δ_{j}, δ_{j})τ)}{θ(y_{α,i} + μ(δ_{i}, δ_{i})τ − y_{β,j} − ν(δ_{j}, δ_{j})τ − (δ_{i}, δ_{j})τ)}

Proposition 4. x is homomorphism of algebras.

Proof. It is direct checking using the formulas (2) and (4).

Remarks. 1. If φ_{α,i} = 0 for some α and i, then f does not contain variable y_{α,i} in corresponding part of the formula (5). For φ_{α,i} > 0, f contains variables y_{α,i} + μ(δ_{i}, δ_{i})τ for 0 ⩽ μ ⩽ φ_{α,i} − 1.

2. It is possible to construct representations of the algebra Q_{n,Δ}(E, τ) using homomorphism x. For this let us construct the representation of the algebra A_{p,Δ}(E, τ) with diagonal action of the elements {y_{α,i}, u_{i}}. The basis of this representation consists of monomials of the elements {e_{α,i}}. The homomorphism x defines the representation of the algebra Q_{n,Δ}(E, τ) in this space.

§3. Intertwining operators.

In this paragraph we assume that Δ is a root system of the finite dimension Lie algebra. The intertwining operators for the representations of the algebra Q_{n,Δ}(E, τ) are constructed in the following way (see also [6], §3). Let σ be an automorphism of the algebra A_{p,Δ}(E, τ). We say that an element C ∈ A_{p,Δ}(E, τ) σ-commutes with the image x(Q_{n,Δ}(E, τ)), if for each element f ∈ Q_{n,Δ}(E, τ) we have: Cx(f) = (σx(f))C in the algebra A_{p,Δ}(E, τ). Let π be some representation of the algebra Q_{n,Δ}(E, τ) constructed by the homomorphism x (see remark 2 in §2). It is clear that C defines the intertwining operator between the representations π and π^{σ} of the algebra Q_{n,Δ}(E, τ). Let C_{1} ∈ A_{p,Δ}(E, τ) σ_{1}-commutes and C_{2} ∈ A_{p,Δ}(E, τ) σ_{2}-commutes with x(Q_{n,Δ}(E, τ)). It is clear that (σ_{2}C_{1})C_{2} σ_{2}σ_{1}-commutes with x(Q_{n,Δ}(E, τ)). This product coincides with the product of the intertwining operators.

Let L_{ℂ} = L ⊗_{ℤ} ℂ be a ℂ-vector space with a basis {δ_{1}, …, δ_{h}}. Let μ ∈ L_{ℂ}, μ = μ_{1}δ_{1} + … + μ_{h}δ_{h}. Let us define an automorphism T_{μ} of the algebra A_{p,Δ}(E, τ) by the following: T_{μ}(e_{α,i}) = e_{α,i}, T_{μ}(y_{α,i}) = y_{α,i}, T_{μ}(u_{i}) = u_{i} +μ_{i}. We will look for the elements of the algebra A_{p,Δ}(E, τ) that T_{μ}-commutes with x(Q_{n,Δ}(E, τ)). The algebra A_{p,Δ}(E, τ) is L-graded, if we assume deg y_{α,i} = deg u_{i} = 0, deg e_{α,i} = δ_{i} for each 1 ⩽ i ⩽ h, 1 ⩽ α ⩽ p_{i}. It is clear that A_{p,Δ}(E, τ) = ⨁_{l∈L} A_{l}, here A_{l} is the

space of elements of degree l and A_{l}A_{l′} ⊂ A_{l+l′} in the algebra A_{p,Δ}(E, τ). It is clear that the homomorphism x preserves the grading, so x(Q_{l}) ⊂ A_{l}. Let ω : L → L_{ℂ} be a homomorphism of abelian groups. We denote by A_{p,Δ}^{ω}(E, τ) the associative algebra such that as a linear space it is isomorphic to A_{p,Δ}(E, τ) and the product ∘ is defined by the following: C_{1} ∘ C_{2} = (T_{ω(l_{2})}C_{1})C_{2}. Here C_{1} ∈ A_{l_{1}}, C_{2} ∈ A_{l_{2}} and ∘ is the product in the algebra A_{p,Δ}^{ω}(E, τ). We will construct a homomorphism

y : Q_{n′,Δ}(E, −τ) → A_{p,Δ}^{ω}(E, τ)

for some n′ and ω such that for each elements f ∈ Q_{n,Δ}(E, τ), g ∈ Q_{n′,Δ}(E, −τ) we will have:

y(g)x(f) = (T_{−ω(l)}x(f))y(g)

in the algebra A_{p,Δ}(E, τ). Here g ∈ Q′_{l}, Q_{n′,Δ}(E, −τ) = ⨁_{l∈L^{+}} Q′_{l} is the decomposi-

tion of Q_{n′,Δ}(E, −τ) for L^{+}-grading. We will have y(Q′_{l}) ⊂ A_{−l}.

Let n′ : L → ℤ be such homomorphism that

n′(δ_{i}) = ∑_{1⩽j⩽h} a_{j,i}p_{j} − n(δ_{i})

We denote n′_{i} = n′(δ_{i}) for all 1 ⩽ i ⩽ h.

Let A′_{p,Δ}(E, −τ) be the algebra with generators {e′_{α,i}, e_{α,i}^{′−1}; 1 ⩽ i ⩽ h, 1 ⩽ α ⩽ p_{i}} and {φ(y′_{1,1}, …, y′_{p_{1},1}; …; y′_{1,h}, …, y′_{p_{h},h}; u′_{1}, …, u′_{h})}, where φ are meromorphic functions. We assume the the relations are as in (4), with τ replaced by −τ and e_{α,i}, y_{α,i}, u_{i} replaced by e′_{α,i}, y′_{α,i}, u′_{i} respectively. It is clear, that there is a homomorphism x′ : Q_{n′,Δ}(E, −τ) → A′_{p,Δ}(E, −τ) defined by (5), where Q_{n,Δ}(E, τ) is replaced by Q_{n′,Δ}(E, −τ) and A_{p,Δ}(E, τ) is replaced by A′_{p,Δ}(E, −τ). Here Q′_{0} consists of meromorphic functions in variables {u′_{1}, …, u′_{h}}.

Let ω : L → L_{ℂ} be such homomorphism that

ω(δ_{i}) = − ∑_{1⩽j⩽h} 2(δ_{i}, δ_{j})τ · δ_{j}

Proposition 5. Following formulas define the homomorphism of the algebras κ : A′_{p,Δ}(E, −τ) → A_{p,Δ}^{ω}(E, τ):

κ(y′_{α,i}) = y_{α,i}
κ(u′_{i}) = −u_{i} − ∑_{\begin{matrix}1⩽j⩽h, \\ 1⩽β⩽p_{j} \end{matrix}} a_{j,i}y_{β,j}  (6)

κ(e′_{α,i}) = ∏_{\begin{matrix}j≠i,1⩽j⩽h, \\ 1⩽β⩽p_{j},0⩽δ⩽−a_{j,i}−1 \end{matrix}}(θ(y_{α,i} − y_{β,j} − ((δ_{i}, δ_{j}) + (δ_{i}, δ_{i}) + δ(δ_{j}, δ_{j}))τ)e^{πiy_{β,j}})×

∏_{\begin{matrix}1⩽β⩽p_{i} \\ β≠α \end{matrix}} \frac{e^{−2πiy_{β,i}}}{θ(y_{α,i} − y_{β,i} − (δ_{i}, δ_{i})τ)θ(y_{α,i} − y_{β,i})} · e_{α,i}^{−1}

Proof. It is a direct checking of the relations (4).

Let T′_{μ} be an automorphism of the algebra A′_{p,Δ}(E, −τ) such that T′_{μ}e′_{α,i} = e′_{α,i}, T′_{μ}y′_{α,i} = y′_{α,i}, T′_{μ}u′_{i} = u′_{i} + μ_{i}. Here μ ∈ L_{ℂ}, μ = μ_{1}δ_{1} + … + μ_{h}δ_{h}. We put:

μ_{i} = (−(n′_{i} + 3)(δ_{i}, δ_{i}) + (δ_{i}, p))τ + \frac{1}{2} ∑_{1⩽j⩽h} a_{j,i}p_{j}

Proposition 6. Let y : Q_{n′,Δ}(E, −τ) → A_{p,Δ}^{ω}(E, τ) be the composition of the homomorphisms x′, T′_{μ} and κ, so y(g) = κT′_{μ}x′(g) for g ∈ Q_{n′,Δ}(E, −τ). Then for each g ∈ Q_{n′,Δ}(E, −τ), f ∈ Q_{n,Δ}(E, τ) there is a relation y(g)x(f) = (T_{−ω(l)}x(f))y(g) in the algebra A_{p,Δ}(E, τ). Here g ∈ Q′_{l}.

Proof. See [6], §3 for the case of strict dominant n and n′ (this means n(δ_{i}) > 0 and n′(δ_{i}) > 0). In general case the proof is similar.

Remarks. 1. This construction for the case of strict dominant n and n′ was done in [6], §3, propositions 3 and 4. We have extended this construction for general case above.

2. In [6], §3, Proposition 5 we have constructed another elements K_{d} ∈ A_{p,Δ}(E, τ) that T_{ν}-commute with x(Q_{n,Δ}(E, τ)). Here d is such that \frac{d}{(δ_{i},δ_{i})} ∈ ℕ for each 1 ⩽ i ⩽ h, ν = −dτ(n′_{1}δ_{1} + … + n′_{h}δ_{h}). In [6] we considered only the case of strict dominant n and n′, but it is clear that the Proposition 5 holds for general case.

3. In [6] we used slightly different notations (compare [6], §1 and §1 of this paper). In notations of [6] we had to divide by n_{i} (see for example [6], formula (1)) but it is not good if we want to consider the case n_{i} = 0 for some i. So we have to change the notations.

§4. Examples and applications.

4.1. Case of finite root system Δ and dominant n.

This case was studied in [6]. We remind the main result about the structure of the algebra Q_{n,Δ}(E, τ).

Let 𝔤 = 𝔤^{−} ⊕ 𝔥 ⊕ 𝔤^{+} be the Cartan decomposition, {g_{γ}; γ ∈ Δ^{+}} be a basis in 𝔤^{+}, φ_{γ_{1},γ_{2}} ∈ ℂ be the structure constants, so [g_{γ_{1}}, g_{γ_{2}}] = φ_{γ_{1},γ_{2}}g_{γ_{1}+γ_{2}}. We denote by 𝔤̂^{+}(E, n) the following L^{+}-graded Lie algebra over field Q_{0}:

𝔤̂^{+}(E, n) = ⨁_{γ∈Δ^{+}} Θ_{n(γ),u(γ)}(Γ)

as a linear space. Here u(γ) = u_{1}γ_{1} + … + u_{h}γ_{h} for γ = γ_{1}δ_{1} + … + γ_{h}δ_{h}. The commutator for μ ∈ Θ_{n(γ),u(γ)}(Γ), λ ∈ Θ_{n(γ′),u(γ′)}(Γ) is by definition [μ, λ] = φ_{γ,γ′}μ·λ. Here μ·λ is the usual product of functions. We remind that by definition Θ_{n(γ),u(γ)}(Γ) = 0 if n(γ) = 0 (see §1.1). It is clear that 𝔤̂^{+}(E, n) is a subalgebra of the current algebra of 𝔤^{+} over field Q_{0}.

Proposition 7. The algebra Q_{n,Δ}(E, τ), defined for finite root system Δ and dominant n is a deformation of the universal enveloping algebra U(𝔤̂^{+}(E, n)) in the class of L^{+}-graded associative algebras.

Remark. From this proposition it follows that the Hilbert function of the algebra Q_{n,Δ}(E, τ) for finite Δ and dominant n is

∑_{l∈L^{+}} dim_{Q_{0}}Q_{l}w^{l} = ∏_{δ∈Δ^{+}}(1 − w^{δ})^{−n(δ)}

Here w^{l} = w_{1}^{l_{1}} … w_{h}^{l_{h}} for l = l_{1}δ_{1} + … + l_{h}δ_{h}; w_{1}, …, w_{h} are formal parameters.

4.2. The Zamolodchikove algebra for the elliptic Belavin R-matrix. Let Δ be root system A_{h}, so (δ_{i}, δ_{i}) = 2, 1 ⩽ i ⩽ h; (δ_{i}, δ_{i+1}) = −1, 1 ⩽ i < h;

(δ_{i}, δ_{j}) = 0 if |i − j| > 1. Let n(δ_{1}) = m > 0, n(δ_{2}) = … = n(δ_{h}) = 0. The Hilbert function of the algebra Q_{n,Δ}(E, τ) has a form (see §4.1): ∏_{γ∈Δ^{+}}(1 − w^{γ})^{−n(γ)} =

(1 − w_{1})^{−m}(1 − w_{1}w_{2})^{−m} … (1 − w_{1} … w_{h})^{−m}. So the algebra Q_{n,Δ}(E, τ) is a deformation of the polynomial ring in hm variables: m variables of degree δ_{1}+…+δ_{i} for each 1 ⩽ i ⩽ h. It is easy to check that the space Q_{δ_{1}+...+δ_{i}} consists of functions of the form:

f(x_{1,1}; …; x_{1,i}; u_{1}, …, u_{h}) = \frac{∏_{1⩽α<i} θ(x_{1,α} − x_{1,α+1} − ∑_{α<ν⩽i} u_{ν} + τ)}{∏_{1⩽α<i} θ(x_{1,α} − x_{1,α+1})}φ(x_{1,1}),

where φ(x) ∈ Θ_{m,u_{1}+...+u_{i}}(Γ).

Proposition 8. The space Q_{δ_{1}+...+δ_{i}} generates the algebra Q_{m}(E, \frac{2}{m}τ) for each 1 ⩽ i ⩽ h. The commutation relations between Q_{δ_{1}+...+δ_{i}} and Q_{δ_{1}+...+δ_{j}}, i ≠ j, 1 ⩽ i, j ⩽ h are defined by the elliptic R-matrix R_{m}(E, \frac{2}{m}τ) (for the definitions of Q_{m}(E, τ) and R_{m}(E, τ) see [1]).

Proof. It is enough to proof the proposition for the image of the homomorphism x : Q_{n,Δ}(E, τ) → A_{p,Δ}(E, τ), because for large {p_{i}} this homomorphism is injection. Let us denote by z_{φ,i} the image of the element from the space Q_{δ_{1}+...+δ_{i}} for a function φ(x) ∈ Θ_{m,u_{1}+...+u_{i}}(Γ). We denote by v_{i} = u_{1} + … + u_{i} for 1 ⩽ i ⩽ h. The formula (5) gives us the following expression for z_{φ,i} ∈ A_{p,Δ}(E, τ):

z_{φ,i} = ∑_{\begin{matrix}1⩽α_{1}⩽p_{1} \\ ... & ... & ... & ... \\ 1⩽α_{i}⩽p_{i} \end{matrix}} \frac{∏_{1⩽ν<i} θ(y_{α_{ν},ν} − y_{α_{ν+1},ν+1} + v_{ν} − v_{i} + τ)}{∏_{1⩽ν<i} θ(y_{α_{ν},ν} − y_{α_{ν+1},ν+1} + τ)}φ(y_{α_{1},1})e_{α_{1},1}e_{α_{2},2} … e_{α_{i},i}

(7)

Let

f_{α_{1},i} = ∑_{\begin{matrix}1⩽α_{2}⩽p_{2} \\ ... & ... & ... & ... \\ 1⩽α_{i}⩽p_{i} \end{matrix}} \frac{∏_{1⩽ν<i} θ(y_{α_{ν},ν} − y_{α_{ν+1},ν+1} + v_{ν} − v_{i} + τ)}{∏_{1⩽ν<i} θ(y_{α_{ν},ν} − y_{α_{ν+1},ν+1} + τ)}e_{α_{1},1}e_{α_{2},2} … e_{α_{i},i} (8)

From (7) and (8) we have:

z_{φ,i} = ∑_{1⩽α⩽p_{1}} φ(y_{α,1})f_{α,i}  (9)

Using (4) it is easy to check the following relations between {f_{α,i}, y_{α,1}, v_{i}; 1 ⩽ α ⩽ p_{1}; 1 ⩽ i ⩽ h}:

f_{α,i}f_{β,j} = \frac{e^{2πiτ} θ(v_{i} − v_{j})θ(y_{α,1} − y_{β,1} − 2τ)}{θ(v_{i} − v_{j} + 2τ)θ(y_{α,1} − y_{β,1})}f_{β,j}f_{α,i}+

\frac{e^{2πi(v_{i}−v_{j}−τ)}θ(2τ)θ(y_{α,1} − y_{β,1} + v_{j} − v_{i})}{θ(v_{i} − v_{j} + 2τ)θ(y_{α,1} − y_{β,1})}f_{α,j}f_{β,i}, here α ≠ β, i < j;  (10)

f_{α,i}f_{β,i} = −e^{2πi(y_{β,1}−y_{α,1})} \frac{θ(y_{α,1} − y_{β,1} − 2τ)}{θ(y_{β,1} − y_{α,1} − 2τ)}f_{β,i}f_{α,i}, here α ≠ β;

f_{α,j}f_{α,i} = e^{2πiτ} f_{α,i}f_{α,j}, here i < j;

f_{α,i}y_{β,1} = y_{β,1}f_{α,i}, here α ≠ β; f_{α,i}y_{α,1} = (y_{α,1} + 2τ)f_{α,i};
f_{α,i}v_{j} = (v_{j} − 2τ)f_{α,i}, here i ≠ j; f_{α,i}v_{i} = (v_{i} − 4τ)f_{α,i};
[y_{α,1}, y_{β,1}] = [y_{α,1}, v_{i}] = [v_{i}, v_{j}] = 0.

The proposition follows from the relations (10), the formula (9) and the results of [2].

4.3. The elliptic deformation of the algebra functions on Grassmannian.

Let us set Δ = A_{h}, (δ_{i}, δ_{i}) = 2, 1 ⩽ i ⩽ h; (δ_{i}, δ_{i+1}) = −1, 1 ⩽ i < h; (δ_{i}, δ_{j}) = 0 if |i − j| > 1 as above. Let n(δ_{1}) = −1, n(δ_{i}) = 0 for 2 ⩽ i ⩽ h − 1, n(δ_{h}) = m > 0. The space Q_{δ_{1}+2δ_{2}+...+hδ_{h}} consists of the following functions:

f(x_{1,1}; x_{1,2}, x_{2,2}; …; x_{1,h}, …, x_{h,h}; u_{1}, …, u_{h}) =

\frac{∏_{1⩽i<h} θ( ∑_{1⩽α⩽i} x_{α,i} − ∑_{1⩽α⩽i+1} x_{α,i+1} + ∑_{1⩽α⩽i} u_{α}) ∏_{\begin{matrix}1⩽i⩽h, \\ 1⩽α≠β⩽i \end{matrix}} θ(x_{α,i} − x_{β,i} − τ)}{∏_{\begin{matrix}1⩽i<h, \\ 1⩽α⩽i,1⩽β⩽i+1 \end{matrix}} θ(x_{α,i} − x_{β,i+1})}× (11)

φ(x_{1,h}, …, x_{h,h}; u_{1}, …, u_{h})

Here φ(x_{1}, …, x_{h}; u_{1}, …, u_{h}) is holomorphic and symmetric in the variables {x_{1}, …, x_{h}} and satisfies relations:

φ(x_{1} + 1, x_{2}, …, x_{h}; u_{1}, …, u_{h}) = φ(x_{1}, …, x_{h}; u_{1}, …, u_{h})
φ(x_{1} + η, x_{2}, …, x_{h}; u_{1}, …, u_{h}) =
e^{−2πi((m−2h+2)x_{1}+x_{2}+...+x_{h}+u_{1}+...+u_{h})}φ(x_{1}, …, x_{h}; u_{1}, …, u_{h})

It is easy to check that the space of such functions is isomorphic to Λ^{h}Θ_{m−h+1,u_{1}+...+u_{h}}(Γ). Proposition 9. For τ = 0 the subalgebra ⨁_{α⩾1} Q_{α(δ_{1}+2δ_{2}+...+hδ_{h})} of the algebra

Q_{n,Δ}(E, τ) is an algebra of functions on the manifold of h-dimensional subspaces in m−h+1-dimensional vector space. This means that the algebra is commutative and generated by Q_{δ_{1}+2δ_{2}+...+hδ_{h}} ≅ Λ^{h}Θ_{m−h+1,u_{1}+...+u_{h}}(Γ) and the Plucker relations. For generic τ the algebra ⨁_{α⩾1} Q_{α(δ_{1}+2δ_{2}+...+hδ_{h})} is a flat deformation of the algebra

of functions on the Grassmannian Gr(h, m − h + 1).

Proof. For τ = 0 the proof is similar to the proof of proposition 8. From our definitions it follows that dim_{Q_{0}}Q_{α(δ_{1}+2δ_{2}+...+hδ_{h})} does not depend on τ. So this dimension is the same as for τ = 0.

4.4. Algebra Q_{hm+1,h}(E, τ).

Let Δ = A_{h}, (δ_{i}, δ_{i}) = 2, 1 ⩽ i ⩽ h; (δ_{i}, δ_{i+1}) = −1, 1 ⩽ i < h; (δ_{i}, δ_{j}) = 0 if |i−j| > 1 as above. Let n(δ_{1}) = 1, n(δ_{i}) = 0 for 2 ⩽ i ⩽ h−1, n(δ_{h}) = m > 0. From the proposition 7 it follows that the algebra Q_{n,Δ}(E, 0) is the algebra of polynomials in h(m + 1) variables: one variable of degree δ_{1} + … + δ_{i} for each 1 ⩽ i < h, m variables of degree δ_{i} + … + δ_{h} for each 1 < i ⩽ h and m + 1 variables of degree δ_{1} + … + δ_{h}. It is easy to see from this that the subalgebra ⨁_{α⩾1} Q_{α(δ_{1}+...+δ_{h})} of

the algebra Q_{n,Δ}(E, 0) is the algebra of polynomials in hm + 1 variables of degree δ_{1} + … + δ_{h}. So the subalgebra ⨁_{α⩾1} Q_{α(δ_{1}+...+δ_{h})} of the algebra Q_{n,Δ}(E, τ) is a flat

deformation of the algebra of polynomials in hm+1 variables. The Hilbert function

is 1 + ∑_{α⩾1} dim_{Q_{0}}Q_{α(δ_{1}+...+δ_{h})}w^{α} = (1 − w)^{−(hm+1)}.

Proposition 10. The subalgebra ⨁_{α⩾1} Q_{α(δ_{1}+...+δ_{h})} of the algebra Q_{n,Δ}(E, τ) is

isomorphic to Q_{hm+1,h}(E, \frac{2}{hm+1}τ).

Proof. The proof is similar to the proof of proposition 8, using the construction of the algebras Q_{n,k}(E, τ) (see [2]).

4.5. Generalized elliptic R-matrix.

Let Δ = A_{h}, (δ_{i}, δ_{i}) = 2, 1 ⩽ i ⩽ h; (δ_{i}, δ_{i+1}) = −1, 1 ⩽ i < h; (δ_{i}, δ_{j}) = 0 if |i − j| > 1 as above. We fix ν, 1 < ν < h. Let n(δ_{ν}) = 1, n(δ_{i}) = 0 for 1 ⩽ i ⩽ h, i ≠ ν. According to the proposition 7 the algebra Q_{n,Δ}(E, 0) is the algebra of polynomials in ν(h − ν + 1) variables: one variable of degree δ_{i_{1}} + δ_{i_{1}+1} + … + δ_{i_{2}} for each i_{1}, i_{2} such that 1 ⩽ i_{1} ⩽ ν ⩽ i_{2} ⩽ h. The algebra Q_{n,Δ}(E, τ) is a flat deformation of this polynomial ring. It is clear that in the algebra Q_{n,Δ}(E, τ) we have dim_{Q_{0}}Q_{δ_{i_{1}} +...+δ_{i_{2}}} = 1 for 1 ⩽ i_{1} ⩽ ν ⩽ i_{2} ⩽ h, Q_{δ_{i_{1}} +...+δ_{i_{2}}} = 0 for ν < i_{1} or i_{2} < ν. Let e_{i_{1},i_{2}} be a nonzero element from Q_{δ_{i_{1}} +δ_{i_{1}+1}+...+δ_{i_{2}}} (for 1 ⩽ i_{1} ⩽ ν ⩽ i_{2} ⩽ h). It is easy to check that we can choose e_{i_{1},i_{2}} in such a form:

e_{i_{1},i_{2}} =

\frac{∏_{i_{1}⩽φ<ν} θ(x_{1,φ} − x_{1,φ+1} + ∑_{i_{1}⩽α⩽φ} u_{α} − τ) ∏_{ν⩽ψ<i_{2}} θ(x_{1,ψ} − x_{1,ψ+1} − ∑_{ψ<α⩽i_{2}} u_{α} + τ)}{∏_{i_{1}⩽p<i_{2}} θ(x_{1,p} − x_{1,p+1})}×

(12)

θ(x_{1,ν} + ∑_{i_{1}⩽α⩽i_{2}} u_{α} − 2τ)

Let us denote v_{i} = u_{i} +u_{i+1} +…+u_{ν} for 1 ⩽ i ⩽ ν and w_{i} = u_{ν} +u_{ν+1} +…+u_{i} for ν ⩽ i ⩽ h. We denote q(j, j′) = e^{2πiτ} for j < j′, q(j, j′) = e^{−2πiτ} for j > j′, q(j, j) = 1.

Proposition 11. The elements {e_{i_{1},i_{2}}, v_{i_{1}}, w_{i_{2}}; 1 ⩽ i_{1} ⩽ ν ⩽ i_{2} ⩽ h} satisfy following commutation relations:

e_{i_{1},i_{2}}e_{i′_{1},i′_{2}} = \frac{θ(v_{i_{1}} − v_{i′_{1}} − 2τ)θ(w_{i′_{2}} − w_{i_{2}})q(i′_{1}, i_{1})q(i′_{2}, i_{2})}{θ(v_{i_{1}} − v_{i′_{1}})θ(w_{i′_{2}} − w_{i_{2}} − 2τ)}e_{i′_{1},i′_{2}}e_{i_{1},i_{2}}+  (13)

\frac{θ(−2τ)q(i′_{2}, i_{2})θ(w_{i′_{2}} − w_{i_{2}} − v_{i_{1}} + v_{i′_{1}})}{θ(v_{i′_{1}} − v_{i_{1}})θ(w_{i′_{2}} − w_{i_{2}} − 2τ)}e_{i_{1},i′_{2}}e_{i′_{1},i_{2}}, for i_{1} ≠ i′_{1}, i_{2} ≠ i′_{2};

e_{i_{1},i_{2}}e_{i′_{1},i_{2}} = q(i′_{1}, i_{1})e_{i′_{1},i_{2}}e_{i_{1},i_{2}}; e_{i_{1},i_{2}}e_{i_{1},i′_{2}} = q(i′_{2}, i_{2})e_{i_{1},i′_{2}}e_{i_{1},i_{2}};

e_{i_{1},i_{2}}v_{i′_{1}} = v_{i′_{1}}e_{i_{1},i_{2}} for i_{1} ≠ i′_{1}; e_{i_{1},i_{2}}w_{i′_{2}} = w_{i′_{2}}e_{i_{1},i_{2}} for i_{2} ≠ i′_{2};

e_{i_{1},i_{2}}v_{i_{1}} = (v_{i_{1}} − 2τ)e_{i_{1},i_{2}}; e_{i_{1},i_{2}}w_{i_{2}} = (w_{i_{2}} − 2τ)e_{i_{1},i_{2}};
[v_{i_{1}}, v_{i′_{1}}] = [w_{i_{2}}, w_{i′_{2}}] = [v_{i_{1}}, w_{i_{2}}] = 0

Proof. It is clear that

e_{i_{1},i_{2}}e_{i′_{1},i′_{2}}, e_{i′_{1},i′_{2}}e_{i_{1},i_{2}}, e_{i_{1},i′_{2}}e_{i′_{1},i_{2}} ∈ Q_{δ_{i_{1}} +δ_{i_{1}+1}+...+δ_{i_{2}} +δ_{i′_{1}} +δ_{i′_{1}+1}+...+δ_{i′_{2}}}

but dim_{Q_{0}}Q_{δ_{i_{1}} +...+δ_{i_{2}} +δ_{i′_{1}} +...+δ_{i′_{2}}} = 2, so there is a linear relation between these elements. It is easy to find the coefficients of this relation using (2). Another relations in (13) are easy to check directly.

Remark. If we put n(δ_{ν}) = m > 0 instead of n(δ_{ν}) = 1, then we will have an algebra that is a deformation of the polynomial ring in mν(h − ν + 1) variables: m variables of degree δ_{i_{1}} + δ_{i_{1}+1} + … + δ_{i_{2}} for each 1 ⩽ i_{1} ⩽ ν ⩽ i_{2} ⩽ h. It is clear that dim_{Q_{0}}Q_{δ_{i_{1}} +...+δ_{i_{2}}} = m for 1 ⩽ i_{1} ⩽ ν ⩽ i_{2} ⩽ h, Q_{δ_{i_{1}} +...+δ_{i_{2}}} = 0 for i_{1} > ν or i_{2} < ν and dim_{Q_{0}}Q_{δ_{i_{1}} +...+δ_{i_{2}} +δ_{i′_{1}} +...+δ_{i′_{2}}} = 2m^{2} for 1 ⩽ i_{1}, i′_{1} ⩽ ν ⩽ i_{2}, i′_{2} ⩽ h. So the elements from the product Q_{δ_{i_{1}} +...+δ_{i_{2}}} ∗ Q_{δ_{i′_{1}} +...+δ_{i′_{2}}} are uniquely expressed as linear combination of the elements from the space Q_{δ_{i′_{1}} +...+δ_{i′_{2}}} ∗ Q_{δ_{i_{1}} +...+δ_{i_{2}}} ⊕ Q_{δ_{i_{1}} +...+δ_{i′_{2}}} ∗ Q_{δ_{i′_{1}} +...+δ_{i_{2}}} . So we have a map

Q_{δ_{i_{1}} +...+δ_{i_{2}}} ⊗ Q_{δ_{i′_{1}} +...+δ_{i′_{2}}} → Q_{δ_{i′_{1}} +...+δ_{i′_{2}}} ⊗ Q_{δ_{i_{1}} +...+δ_{i_{2}}} ⊕ Q_{δ_{i_{1}} +...+δ_{i′_{2}}} ⊗ Q_{δ_{i′_{1}} +...+δ_{i_{2}}}

It is clear that this map satisfies the Yang-Baxter equations.

§5. Case of affine root system. A family of commuting elements.

Let Δ be the root system of the Lie algebra ŝl_{h}. So (δ_{i}, δ_{i}) = 2, (δ_{i}, δ_{i+1}) = −1 and (δ_{i}, δ_{j}) = 0 otherwise. Here we assume that i ∈ ℤ_{h}. It is possible to prove that for dominant n the Hilbert function of Q_{n,Δ}(E, τ) is equal to

∏_{γ∈Δ^{+}}(1 − w^{γ})^{−n(γ)k(γ)} ∏_{α⩾μ}(1 − w^{α(δ_{1}+...+δ_{h})})^{−αn(δ_{1}+...+δ_{h})}

Here k(γ) is a multiplicity of the root γ in the algebra ŝl_{h}, μ = 2 for h = 2 and μ = 1 for h > 2. So the algebra Q_{n,Δ}(E, τ) is bigger then in the finite dimensional case. Note that the element u_{1} + … + u_{h} is central in the algebra Q_{n,Δ}(E, τ), because the root δ_{1} + … + δ_{h} is imaginary. We denote by Q_{n,Δ,c}(E, τ) (for c ∈ ℂ) the factor algebra of Q_{n,Δ}(E, τ) by the relation u_{1} + … + u_{h} = c. If n is dominant and n ≠ 0 then the Hilbert function of the algebra Q_{n,Δ,c}(E, τ) is the same as for the algebra Q_{n,Δ}(E, τ) and does not depend on c. Let us n = 0. The Hilbert function of the algebra Q_{0,Δ,c}(E, τ) is equal to 1 if c ∉ Γ and is equal to ∏_{α⩾1}(1 − w^{α(δ_{1}+...+δ_{h})})^{−h}

for h > 2 and (1 − w_{1}w_{2})^{−1} ∏_{α⩾2}(1 − w_{1}^{α}w_{2}^{α})^{−2} for h = 2 if c ∈ Γ.

Proposition 12. Let c ∈ Γ. The algebra Q_{0,Δ,c}(E, τ) is commutative for each τ ∈ E and isomorphic to polynomials in infinite number of variables. Namely, for h > 2 it is the algebra of polynomials in h variables of degree α(δ_{1} + … + δ_{h}) for each α ⩾ 1 and for h = 2 it is the algebra of polynomials in one variable of degree δ_{1} + δ_{2} and two variables of degree α(δ_{1} + δ_{2}) for each α ⩾ 2.

Proof. Let c = 0. Firstly we consider the case h > 2. Let g(z_{1}, …, z_{h}) be the holomorphic function in variables {z_{j}; j ∈ ℤ_{h}} that satisfies the relations:

g(z_{1}, …, z_{j} + 1, …, z_{h}) = g(z_{1}, …, z_{h}),

g(z_{1}, …, z_{j} +η, …, z_{h}) = e^{−2πi(2z_{j}−z_{j−1}−z_{j+1}−η+u_{j})}g(z_{1}, …, z_{h}) for each j ∈ ℤ_{h}. It is easy to see that the space of such functions is h-dimensional (over field of functions in variables {u_{j}}) and g(z_{1} + p, …, z_{h} + p) = g(z_{1}, …, z_{h}) for each p ∈ ℂ. Let K_{α,g} be the following element of the space Q_{α(δ_{1}+...+δ_{h})}:

K_{α,g} = \frac{∏_{\begin{matrix}i∈ℤ_{h}, \\ 1⩽μ,ν⩽α \end{matrix}} θ(x_{μ,i} − x_{ν,i} − 2τ) · g(x_{1,1} + … + x_{α,1}, …, x_{1,h} + … + x_{α,h})}{∏_{\begin{matrix}i∈ℤ_{h}, \\ 1⩽μ,ν⩽α \end{matrix}} θ(x_{μ,i} − x_{ν,i+1})}

(14)

One can check by direct calculation that K_{α_{1},g_{1}} ∗ K_{α_{2},g_{2}} = K_{α_{2},g_{2}} ∗ K_{α_{1},g_{1}} for each α_{1}, α_{2}; g_{1}, g_{2}.

For h = 2 the elements K_{α,g} defined by (14) commute also, but in this case K_{α,g} ∉ Q_{α(δ_{1}+δ_{2})} because K_{α,g} has a pole of order 2. In this case the algebra Q_{0,Δ,0}(E, τ) is a subalgebra of the algebra generated by {K_{α,g}}.

§6. Elliptic deformation of the Poison algebra. Another family of commuting elements.

Let τ_{1}, τ_{2}, τ_{3} ∈ E, τ_{1} + τ_{2} + τ_{3} = 0 and m ∈ ℕ. We define an associative ℕ-graded algebra H_{m}(E, τ_{1}, τ_{2}, τ_{3}) by the following: as a linear space H_{m}(E, τ_{1}, τ_{2}, τ_{3}) = F_{1}⊕F_{2}⊕F_{3}⊕…, where F_{α}∗F_{β} ⊂ F_{α+β} and F_{α} is a space of functions f(x_{1}, …, x_{α}) on ℂ^{α} that satisfies the properties:

1. f(x_{1}, …, x_{α}) is symmetric by x_{1}, …, x_{α}.

2. f(x_{1}, …, x_{α}) is holomorphic outside the divisors {x_{μ} − x_{ν} = 0} and has a pole of order ⩽ 2 on these divisors.

3. For α ⩾ 3 we have:

f(x, x + τ_{1}, x + τ_{1} + τ_{2}, x_{4}, …, x_{α}) = 0
f(x, x + τ_{2}, x + τ_{2} + τ_{1}, x_{4}, …, x_{α}) = 0

4. f(x_{1} + 1, x_{2}, …, x_{α}) = f(x_{1}, …, x_{α}),

f(x_{1} + η, x_{2}, …, x_{α}) = e^{−2πi(mx_{1}+c)}f(x_{1}, …, x_{α}), here c ∈ ℂ is fixed.

We define the product ∗ in the algebra H_{m}(E, τ_{1}, τ_{2}, τ_{3}) by the following rule: for f ∈ F_{α}, g ∈ F_{β} we set:

f ∗ g(x_{1}, …, x_{α+β}) =

\frac{1}{α!β!} ∑_{σ∈S_{α+β}} f(x_{σ_{1}}, …, x_{σ_{α}})g(x_{σ_{α+1}}, …, x_{σ_{α+β}}) ∏_{\begin{matrix}1⩽μ⩽α, \\ α+1⩽ν⩽α+β \end{matrix}} λ(x_{σ_{μ}}, x_{σ_{ν}}),

here λ(x, y) = \frac{θ(x−y−τ_{1})θ(x−y−τ_{2})θ(x−y−τ_{3})}{θ(x−y)^{3}}.

It is clear that for m > 0 the algebra H_{m}(E, τ_{1}, τ_{2}, τ_{3}) does not depend on c. It is possible to proof that for m > 0 the algebra H_{m}(E, τ_{1}, τ_{2}, τ_{3}) is a deformation in class of ℕ-graded associative algebras of the universal enveloping algebra of the following ℕ-graded Lie algebra h_{m,c}(E): as a vector space, h_{m,c}(E) = Θ_{m,c}(Γ) ⊕ Θ_{2m,2c}(Γ) ⊕ … and for φ ∈ Θ_{αm,αc}(Γ), ψ ∈ Θ_{βm,βc}(Γ), the commutator [φ, ψ] = βφ′ψ − αψ′φ ∈ Θ_{(α+β)m,(α+β)c}(Γ). The Hilbert function of the algebra

H_{m}(E, τ_{1}, τ_{2}, τ_{3}) is 1 + ∑_{α⩾1} dim F_{α}w^{α} = ∏_{α⩾1}(1 − w^{α})^{−mα} (see also [5]).

Now let us apply this construction to the case m = 0. For c ∉ Γ we have H_{0}(E, τ_{1}, τ_{2}, τ_{3}) = 0. For c ∈ Γ the Hilbert function of the algebra H_{0}(E, τ_{1}, τ_{2}, τ_{3}) is equal to ∏_{α⩾1}(1 − w^{α})^{−1}. In this case the algebra H_{0}(E, τ_{1}, τ_{2}, τ_{3}) is commutative

for each τ_{1}, τ_{2}, τ_{3} ∈ ℂ, τ_{1} + τ_{2} + τ_{3} = 0 and isomorphic to the polynomial algebra in infinite number of variables: one variable of degree α for each α ⩾ 1. Let us c = 0. We define the elements K_{α} ∈ F_{α} by the formula:

K_{α}(x_{1}, …, x_{α}) = ∏_{1⩽μ<ν⩽α} \frac{θ(x_{μ} − x_{ν} − τ_{1})θ(x_{μ} − x_{ν} + τ_{1})}{θ(x_{μ} − x_{ν})^{2}} for α > 1,

K_{1}(x_{1}) = 1

It is easy to check by direct calculation that K_{α} ∗ K_{β} = K_{β} ∗ K_{α} for each α, β ∈ ℕ (see for details [5]).

References.

1. A.V.Odesskii and B.L.Feigin, ”Sklyanin’s elliptic algebras”, Funkts. Anal. Philozhen., 23, No. 3, 45-54 (1989).

2. A.V.Odesskii and B.L.Feigin, ”Constructions of Sklyanin elliptic algebras and quantum R-matrices”, Funkts. Anal. Philozhen. 27, No. 1, 37-45 (1993).

3. B.L.Feigin and A.V.Odesskii, Vector Bundles on Elliptic Curve and Sklyanin Algebras, RIMS-1032, September 1995, Kyoto University, Kyoto, Japan.

4. Cherednik I.V. On R-matrix quantization of formal loop groups. Group theoretical methods in physics, Voll. 2 (Yurmala, 1985), 161-180, VNU Sci. Press, Utrecht, 1986.

5. B.L.Feigin and A.V.Odesskii, A family of elliptic algebras. Internat. Math. Res. Notices 1997, No 11, 531-539.

6. A.V.Odesskii and B.L.Feigin, Elliptic deformation of current algebras and their representations by difference operators. Funct. Anal. Appl. 31 (1997), No 3, 193-203 (1998).