Functional realization of some elliptic

Hamiltonian structures and bosonization

of the corresponding quantum algebras

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

Introduction

Let 𝔭 be the parabolic subalgebra of some semisimple Lie algebra 𝔤 and P the corresponding group. Let M(E, 𝔭) be the moduli space of P -bundles on the elliptic curve E. In [5] we define the Hamiltonian structure on the manifold M(E, 𝔭). There is a natural problem: to quantize the coordinate ring of each connected component of M(E, 𝔭).

We denote by Q_{n}(E, τ) the corresponding quantum algebras in the case 𝔤 = sl_{2}. Here τ ∈ E is a parameter of quantization, n ∈ ℕ is a number of the connected component of M(E, 𝔭). This component is isomorphic to ℙ^{n−1} in this case. So its coordinate ring is isomorphic to the polynomial ring in n variables and the algebra Q_{n}(E, τ) is a graded deformation of this polynomial ring. We denote the corresponding Poisson algebra by q_{n}(E).

More generally, we denote by Q_{n,k}(E, τ) the corresponding quantum algebras in the case 𝔤 = sl_{k+1} and 𝔭 is a parabolic subalgebra for the flag 0 ⊂ V ⊂ ℂ^{k+1}, dimV = 1. Here τ ∈ E is a parameter of quantization, n ∈ ℕ is a number of the connected component of M(E, 𝔭). If n and k have no common divisors, then this component is isomorphic to ℙ^{n−1}. So its coordinate ring is isomorphic to the polynomial ring in n variables and the algebra Q_{n,k}(E, τ) is a graded deformation of this polynomial ring. We denote the corresponding Poisson algebra by q_{n,k}(E). We have Q_{n}(E, τ) = Q_{n,1}(E, τ), q_{n}(E) = q_{n,1}(E).

In the papers [1,2] we constructed the family of associative algebras Q_{n}(E, τ). The algebra Q_{n}(E, τ) is ℤ-graded and depends on 2 continuous parameters: an elliptic curve E = ℂ⧸Γ and a point τ ∈ E. We have Q_{n}(E, τ) = ℂ ⊕ F_{1} ⊕ F_{2} ⊕ … and F_{α} ∗ F_{β} ⊂ F_{α+β}. The Hilbert function is 1 + ∑_{α⩾1} dim F_{α}t^{α} = (1 − t)^{−n}. If

τ = 0 then the product ∗ is commutative and the algebra Q_{n}(E, 0) is a polynomial ring in n variables. So the algebra Q_{n}(E, 0) does not depend on E. Considering the product ∗ in a neighborhood of τ = 0 with the fixed curve E we will have a Poisson structure on the polynomial ring Q_{n}(E, 0) that depends on E. We call the following construction of the algebra Q_{n}(E, τ) its functional realization, because the graded components F_{α} are described as the spaces of functions and the product ∗ is given by an explicit formula. Namely, F_{α} is a space of holomorphic symmetric functions in α variables f(z_{1}, …, z_{α}) with the properties: f(z_{1} + 1, z_{2}, …, z_{α}) = f(z_{1}, …, z_{α}), f(z_{1} + η, z_{2}, …, z_{α}) = e^{−2πi(nz_{1}−αnτ)}f(z_{1}, …, z_{α}). Here 1, η are generators of the lattice Γ, Im η > 0. It is clear that F_{α} ≅ S^{α}(Θ_{n,−αnτ} (Γ)) (see Notations) and dim F_{α} = \frac{n(n+1)...(n+α−1)}{α!}. The product ∗ is given in the following way: for f ∈ F_{α}, g ∈ F_{β} we have

f ∗ g(z_{1}, …, z_{α+β}) =  (1)

\frac{1}{α!β!} ∑_{σ∈S_{α+β}} f(z_{σ_{1}}, …, z_{σ_{α}})g(z_{σ_{α+1}}−2ατ, …, z_{σ_{α+β}}−2ατ) ∏_{\begin{matrix}1⩽p⩽α \\ α+1⩽q⩽α+β \end{matrix}} \frac{θ(z_{σ_{p}} − z_{σ_{q}} − nτ)}{θ(z_{σ_{p}} − z_{σ_{q}})}

If τ = 0 then the formula (1) gives the usual product in the symmetric algebra S^{∗}(Θ_{n,0}(Γ)). Decompose the right side of the formula (1) in the Taylor series f ∗ g = fg + c_{1}(f, g)τ + o(τ), where fg is the product in the symmetric algebra S^{∗}(Θ_{n,0}(Γ)). Then we will have the explicit formula for the Poisson structure on the algebra S^{∗}(Θ_{n,0}(Γ)). As usual, {f, g} = c_{1}(f, g)−c_{1}(g, f). In the explicit form:

{f, g}(z_{1}, …, z_{α+β}) =

\frac{1}{α!β!} ∑_{σ∈S_{α+β}} ( − 2nf(z_{σ_{1}}, …, z_{σ_{α}})g(z_{σ_{α+1}}, …, z_{σ_{α+β}}) ∑_{\begin{matrix}1⩽p⩽α \\ α+1⩽q⩽α+β \end{matrix}} \frac{θ′(z_{σ_{p}} − z_{σ_{q}})}{θ(z_{σ_{p}} − z_{σ_{q}})}+

+2βg(z_{σ_{α+1}}, …, z_{σ_{α+β}}) ∑_{1⩽p⩽α} f′_{z_{σp}} (z_{σ_{1}}, …, z_{σ_{α}})−

−2αf(z_{σ_{1}}, …, z_{σ_{α}}) ∑_{α+1⩽q⩽α+β} g′_{z_{σq}} (z_{σ_{α+1}}, …, z_{σ_{α+β}}))

There is another construction of the algebra Q_{n}(E, τ) that we call bosonization. Let A_{p}(E, τ) be the algebra generated by {e_{1}, …, e_{p}, φ(z_{1}, …, z_{p})}, where φ is any meromorphic functions in variables z_{1}, …, z_{p}. We assume that the following relations hold:

z_{α}z_{β} = z_{β}z_{α}, e_{α}z_{β} = (z_{β} − 2τ)e_{α}, e_{α}z_{α} = (z_{α} + (n − 2)τ)e_{α}

e_{α}e_{β} = −e^{2πi(z_{β}−z_{α})} \frac{θ(z_{α} − z_{β} − nτ)}{θ(z_{β} − z_{α} − nτ)}e_{β}e_{α}, here α ≠ β

There is a homomorphism x : Q_{n}(E, τ) → A_{p}(E, τ) such that for f ∈ F_{1} = Θ_{n,−nτ} (Γ) we have: x(f) = ∑_{1⩽α⩽p} f(z_{α})e_{α}. It is clear that for τ = 0 the alge-

bra A_{p}(E, 0) is commutative and x is the homomorphism of the Poisson algebras.

It is easy to see that the functional realization of the algebra Q_{n}(E, τ) (and the corresponding Poisson algebra) follows from the bosonization.

In [1,2] we studied the algebras Q_{n,k}(E, τ). Here 1 ⩽ k < n; n and k have no common divisors. We have Q_{n}(E, τ) = Q_{n,1}(E, τ). In the general case the algebra Q_{n,k}(E, τ) is also ℤ-graded, so we have: Q_{n,k}(E, τ) = ℂ ⊕ P_{1} ⊕ P_{2} ⊕ … and P_{α} ∗ P_{β} ⊂ P_{α+β}. The Hilbert function is 1 + ∑_{α⩾1} dim P_{α}t^{α} = (1 − t)^{−n}. If

τ = 0 than the algebra Q_{n,k}(E, 0) is the polynomial ring in n variables. We denote the corresponding Poisson algebra by q_{n,k}(E). Let \frac{n}{k} = n_{1} − _{\frac{1}{n_{2}−···− \frac{1}{np}}} be the

decomposition into the continuous fraction, where n_{1}, …, n_{p} ⩾ 2. It is clear that n = d(n_{1}, …, n_{p}), k = d(n_{2}, …, n_{p}). The space P_{1} is isomorphic to the space Θ_{(n_{1},...,n_{p})}(Γ) (see Notations). We have dim Θ_{(n_{1},...,n_{p})}(Γ) = n.

In [3,4] we consider the case when 𝔤 is general and 𝔭 is a Borel subalgebra. We denote the corresponding quantum algebras by Q_{n,Δ}(E, τ). Here Δ is the root system of 𝔤, n : L → ℤ is the homomorphism of the additive groups, L is the lattice generated by Δ.

In this paper we consider the case 𝔤 = sl_{N} and a general parabolic subalgebra 𝔭. We introduce a functional realization of the Poisson algebras in this case. We also introduce a certain construction of the corresponding quantum algebras. We denote these quantum algebras by Q_{n_{1},k_{1}} ˆ⊗ … ˆ⊗Q_{n_{h},k_{h}}(E, τ) and the corresponding Poisson algebras by q_{n_{1},k_{1}} ˆ⊗ … ˆ⊗q_{n_{h},k_{h}}(E).

Now we describe the contents of the paper.

In §1 we construct the functional realization of the Poisson algebra q_{n,k}(E). In this construction P_{α} = S^{α}(Θ_{(n_{1},...,n_{p})}(Γ)) is realized as the space of holomorphic functions f(x_{1,1}, …, x_{p,1}; …; x_{1,α}, …, x_{p,α}) satisfying some properties. The Poisson bracket is given by the formulas (2) and (3).

In §2 we construct an analogue of bosonization of the algebras Q_{n,k}(E, τ).

In §§3-5 we construct the algebras that we denote by Q_{n_{1},k_{1}} ˆ⊗ … ˆ⊗Q_{n_{h},k_{h}}(E, τ) and the corresponding Poisson algebras q_{n_{1},k_{1}} ˆ⊗ … ˆ⊗q_{n_{h},k_{h}}(E). These algebras

are ℤ^{h}-graded, so Q_{n_{1},k_{1}} ˆ⊗ … ˆ⊗Q_{n_{h},k_{h}}(E, τ) = ⨁_{α_{1},...,α_{h}⩾0} P_{α_{1},...,α_{h}} and P_{α_{1},...,α_{h}} ∗

P_{β_{1},...,β_{h}} ⊂ P_{α_{1}+β_{1},...,α_{h}+β_{h}}. The space M = P_{0,...,0} is the field of meromorphic functions in variables z_{1,2}, …, z_{h−1,h}. The dimensions dim_{M}P_{α_{1},...,α_{h}} are finite and the Hilbert function is

∑_{α_{1},...,α_{h}⩾0} dim_{M}P_{α_{1},...,α_{h}}t_{1}^{α_{1}} … t_{h}^{α_{h}} = ∏_{1⩽λ⩽ν⩽h}(1 − t_{λ}t_{λ+1} … t_{ν})^{−d(N_{λ}ΔN_{λ+1}Δ...ΔN_{ν})}

Here N_{α} = (n_{1,α}, n_{2,α}, …, n_{p_{α},α}), where \frac{n_{α}}{k_{α}} = n_{1,α} − _{\frac{1}{n_{2,α}−···− \frac{1}{np_{α},α}}} is the decom-

position into the continuous fraction, 1 ⩽ α ⩽ h, n_{i,j} ⩾ 2. See Notations for the definition Δ.

In §3 we construct the functional realization of the Poisson algebra q_{n,k} ˆ⊗q_{m,l}(E). In §4 we construct the functional realization of the Poisson algebra q_{n_{1},k_{1}} ˆ⊗ … ˆ⊗q_{n_{h},k_{h}}(E) in general case.

In §5 we construct the bosonization of the algebra Q_{n_{1},k_{1}} ˆ⊗ … ˆ⊗Q_{n_{h},k_{h}}(E, τ) in general case.

In [3,4] we consider the case k_{1} = ⋯ = k_{h} = 1. In this case 𝔤 = sl_{h+1} and 𝔭 is a Borel subalgebra. In those papers the algebra Q_{n_{1},1} ˆ⊗ … ˆ⊗Q_{n_{h},1}(E, τ) was denoted by Q_{n,Δ}(E, τ). Here Δ is the root system A_{h}, n : L → ℤ is the homomorphism of the additive groups, L is the lattice generated by Δ, n_{α} = n(δ_{α}) where δ_{1}, …, δ_{h} are simple positive roots. In [3] §1 and [4] §1 we constructed the functional realization of this algebra. In [3] §1 and [4] §2 we constructed the bosonization of this algebra.

Notations

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 Θ_{m,c}(Γ) the space of holomorphic functions f(z) with the following properties: f(z+1) = f(z), f(z+η) = e^{−2πi(mz+c)}f(z). It is clear that dim Θ_{m,c}(Γ) = m if m > 0. For m > 0 the elements of Θ_{m,c}(Γ) are called θ-functions of order m. 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).

For the sequence of natural numbers (n_{1}, …, n_{p}) we denote d(n_{1}, …, n_{p}) = det (m_{i,j}), here (m_{i,j}) is the p × p matrix with the elements m_{i,i} = n_{i}, m_{i,i+1} = m_{i+1,i} = −1, m_{i,j} = 0 for |i − j| > 1. For p = 0 we assume d(∅) = 1. Let n_{i} ⩾ 2 for all 1 ⩽ i ⩽ p, n = d(n_{1}, …, n_{p}), k = d(n_{2}, …, n_{p}). It is clear that n and k have no common divisors, 1 ⩽ k < n and \frac{n}{k} = n_{1} − _{\frac{1}{n_{2}−_{\frac{1}{n_{3}−···− \frac{1}{np}}}}} .

For two sequences A = (a_{1}, …, a_{p}) and B = (b_{1}, …, b_{q}) we denote AΔB = (a_{1}, …, a_{p−1}, a_{p} + b_{1}, b_{2}, …, b_{q}). It is a sequence of length p + q − 1. Particularly, for p = q = 1 we have AΔB = (a_{1} + b_{1}). It is clear that the operation Δ is associative.

We denote by Θ_{(n_{1},...,n_{p})}(Γ) the space of holomorphic functions f(z_{1}, …, z_{p}) with the following properties:

f(z_{1}, …, z_{α} + 1, …, z_{p}) = f(z_{1}, …, z_{p})
f(z_{1}, …, z_{α} + η, …, z_{p}) = e^{−2πi(n_{α}z_{α}−z_{α−1}−z_{α+1})}f(z_{1}, …, z_{p})

We assume here z_{0} = z_{p+1} = 0. It is easy to check that dim Θ_{(n_{1},...,n_{p})}(Γ) = d(n_{1}, …, n_{p}) for n_{1}, …, n_{p} ⩾ 2.

Let N = (n_{1}, …, n_{p}), M = (m_{1}, …, m_{q}) be two sequences of natural numbers. Let f(z_{1}, …, z_{p}) ∈ Θ_{(n_{1},...,n_{p})}(Γ), g(z_{1}, …, z_{q}) ∈ Θ_{(m_{1},...,m_{q})}(Γ). It is clear that if φ(z_{1}, …, z_{p+q−1}) = f(z_{1}, …, z_{p})g(z_{p}, …, z_{p+q−1}) then φ ∈ Θ_{NΔM}(Γ).

Let A be some commutative associative algebra. We remind that the structure of a Lie algebra {, } on the space A with the following property (Leibniz rule): {f, gh} = {f, g}h + g{f, h} for f, g, h ∈ A is called a Poisson structure on A. The algebra A with a Poisson structure is called a Poisson algebra. Let ∗_{t} be such a family of associative products on the space A that holomorphically depends on t ∈ U ⊂ ℂ, here U is an open subset, 0 ∈ U, and f ∗_{t} g = fg + \frac{1}{2}\{f, g\}t + o(t) as t → 0. Let A_{t} be the associative algebra with the product ∗_{t}. The family of associative algebras A_{t} is called a quantization of the Poisson algebra A.

§1. Functional realization of the Poisson algebra q_{n,k}(E)

Let q_{n,k}(E) = S^{∗}(Θ_{(n_{1},...,n_{p})}(Γ)), where \frac{n}{k} = n_{1} −_{\frac{1}{n_{2}−···− \frac{1}{np}}} is the decomposition

into the continuous fraction, n_{1}, …, n_{p} ⩾ 2. So q_{n,k}(E) is the polynomial ring in n variables and the space S^{α}(Θ_{(n_{1},...,n_{p})}(Γ)) of elements of degree α is realized as the space of holomorphic functions f(x_{1,1}, …, x_{p,1}; …; x_{1,α}, …, x_{p,α}) with the following properties:

1. Symmetry. For each σ ∈ S_{α} we have f(x_{1,σ_{1}}, …, x_{p,σ_{1}}; …; x_{1,σ_{α}}, …, x_{p,σ_{α}}) = f(x_{1,1}, …, x_{p,1}; …; x_{1,α}, …, x_{p,α}).

2. Periodicity and quasiperiodicity.

f(x_{1,1}, …, x_{μ,ν} + 1, …, x_{p,α}) = f(x_{1,1}, …, x_{p,α})

f(x_{1,1}, …, x_{μ,ν} + η, …, x_{p,α}) = e^{−2πi(n_{μ}x_{μ,ν}−x_{μ−1,ν}−x_{μ+1,ν})}f(x_{1,1}, …, x_{p,α})

We assume here x_{0,ν} = x_{p+1,ν} = 0.

The product on the space q_{n,k}(E) is given by the usual formula:

for f ∈ S^{α}(Θ_{(n_{1},...,n_{p})}(Γ)), g ∈ S^{β}(Θ_{(n_{1},...,n_{p})}(Γ)) we have:

fg(x_{1,1}, …, x_{p,1}; …; x_{1,α+β}, …, x_{p,α+β}) = \frac{1}{α!β!} ∑_{σ∈S_{α+β}}

f(x_{1,σ_{1}}, …, x_{p,σ_{1}}; …; x_{1,σ_{α}}, …, x_{p,σ_{α}})g(x_{1,σ_{α+1}}, …, x_{p,σ_{α+1}}; …; x_{1,σ_{α+β}}, …, x_{p,σ_{α+β}})

We define the operation {, } on the space q_{n,k}(E) in the following way: if f, g ∈ Θ_{(n_{1},...,n_{p})}(Γ) then for {f, g} ∈ S^{2}(Θ_{(n_{1},...,n_{p})}(Γ)) we have:

{f, g}(x_{1}, …, x_{p}; y_{1}, …, y_{p}) =  (2)

∑_{1⩽α⩽p} \frac{d(n_{1}, …, n_{α−1}) + d(n_{α+1}, …, n_{p})}{d(n_{1}, …, n_{p})}(g(x_{1}, …, x_{p})f′_{y_{α}}(y_{1}, …, y_{p})+

+g(y_{1}, …, y_{p})f′_{x_{α}}(x_{1}, …, x_{p})−f(x_{1}, …, x_{p})g′_{y_{α}}(y_{1}, …, y_{p})−f(y_{1}, …, y_{p})g′_{x_{α}}(x_{1}, …, x_{p}))+

+(\frac{θ′(y_{1} − x_{1})}{θ(y_{1} − x_{1})}+\frac{θ′(y_{p} − x_{p})}{θ(y_{p} − x_{p})}−2πi)(f(x_{1}, …, x_{p})g(y_{1}, …, y_{p})−g(x_{1}, …, x_{p})f(y_{1}, …, y_{p}))+

θ′(0) ∑_{1⩽α<p} \frac{θ(x_{α} + y_{α+1} − y_{α} − x_{α+1})}{θ(x_{α} − y_{α})θ(y_{α+1} − x_{α+1})}(f(y_{1}, …, y_{α}, x_{α+1}, …, x_{p})g(x_{1}, …, x_{α}, y_{α+1}, …, y_{p})−

−g(y_{1}, …, y_{α}, x_{α+1}, …, x_{p})f(x_{1}, …, x_{α}, y_{α+1}, …, y_{p}))

In the general case, if f ∈ S^{α}(Θ_{(n_{1},...,n_{p})}(Γ)), g ∈ S^{β}(Θ_{(n_{1},...,n_{p})}(Γ)) then for {f, g} ∈ S^{α+β}(Θ_{(n_{1},...,n_{p})}(Γ)) we have by definition:

\{f, g\}(x_{1,1}, …, x_{p,1}; …; x_{1,α+β}, …, x_{p,α+β}) = \frac{1}{α!β!} ∑_{σ∈S_{α+β}}  (3)

(βg(x_{1,σ_{α+1}}, …, x_{p,σ_{α+β}}) ∑_{\begin{matrix}1⩽ψ⩽p \\ 1⩽μ⩽α \end{matrix}} \frac{d(n_{1}, …, n_{ψ−1}) + d(n_{ψ+1}, …, n_{p})}{d(n_{1}, …, n_{p})}f′_{x_{ψ,σμ}} (x_{1,σ_{1}}, …, x_{p,σ_{α}})−

αf(x_{1,σ_{1}}, …, x_{p,σ_{α}}) ∑_{\begin{matrix}1⩽ψ⩽p \\ α+1⩽μ⩽α+β \end{matrix}} \frac{d(n_{1}, …, n_{ψ−1}) + d(n_{ψ+1}, …, n_{p})}{d(n_{1}, …, n_{p})}g′_{x_{ψ,σμ}} (x_{1,σ_{α+1}}, …, x_{p,σ_{α+β}})+

+( ∑_{\begin{matrix}1⩽μ⩽α \\ α+1⩽μ′⩽α+β \end{matrix}} \frac{θ′(x_{1,σ_{μ′}} − x_{1,σ_{μ}})}{θ(x_{1,σ_{μ′}} − x_{1,σ_{μ}})} + \frac{θ′(x_{p,σ_{μ′}} − x_{p,σ_{μ}})}{θ(x_{p,σ_{μ′}} − x_{p,σ_{μ}})} − 2πiαβ)×

×f(x_{1,σ_{1}}, …, x_{p,σ_{α}})g(x_{1,σ_{α+1}}, …, x_{p,σ_{α+β}})+

+θ′(0) ∑_{\begin{matrix}1⩽μ⩽α \\ α+1⩽μ′⩽α+β \\ 1⩽ψ⩽p−1 \end{matrix}} \frac{θ(x_{ψ,σ_{μ}} + x_{ψ+1,σ_{μ′}} − x_{ψ,σ_{μ′}} − x_{ψ+1,σ_{μ}})}{θ(x_{ψ,σ_{μ}} − x_{ψ,σ_{μ′}})θ(x_{ψ+1,σ_{μ′}} − x_{ψ+1,σ_{μ}})}×

×f(x_{1,σ_{1}}, …, x_{p,σ_{1}}; …; x_{1,σ_{μ′}} , …, x_{ψ,σ_{μ′}} , x_{ψ+1,σ_{μ}}, …, x_{p,σ_{μ}}; …; x_{1,σ_{α}}, …, x_{p,σ_{α}})× ×g(x_{1,σ_{α+1}}, …, x_{p,σ_{α+1}}; …; x_{1,σ_{μ}}, …, x_{ψ,σ_{μ}}, x_{ψ+1,σ_{μ′}} , …, x_{p,σ_{μ′}}; …; x_{1,σ_{α+β}}, …, x_{p,σ_{α+β}}))

Proposition 1. The operation {, } defines a Poisson bracket on the space q_{n,k}(E).

§2. Bosonization of the algebra Q_{n,k}(E, τ)

Let \frac{n}{k} = n_{1} − _{\frac{1}{n_{2}−···− \frac{1}{np}}} , n_{1}, …, n_{p} ⩾ 2. Let A_{m_{1},...,m_{p}}(E, τ) be the associative

algebra generated by {e_{α_{1},...,α_{p}}; 1 ⩽ α_{t} ⩽ m_{t}, 1 ⩽ t ⩽ p} and {φ(y_{1,1}, …, y_{p,m_{p}})}, where φ is any meromorphic function in variables {y_{λ,α}; 1 ⩽ λ ⩽ p, 1 ⩽ α ⩽ m_{λ}}. We assume that the following relations hold (see also [2] for the cases p = 1 and p = 2):

y_{λ,α}y_{ν,β} = y_{ν,β}y_{λ,α},

e_{α_{1},...,α_{p}}y_{ν,β} = (y_{ν,β} − (d(n_{1}, …, n_{ν−1}) + d(n_{ν+1}, …, n_{p}))τ)e_{α_{1},...,α_{p}}, here α_{ν} ≠ β

e_{α_{1},...,α_{p}}y_{ν,α_{ν}} = (y_{ν,α_{ν}} +(d(n_{1}, …, n_{p})−d(n_{1}, …, n_{ν−1})−d(n_{ν+1}, …, n_{p}))τ)e_{α_{1},...,α_{p}}

e_{α_{1},...,α_{p}}e_{β_{1},...,β_{p}} = Λe_{β_{1},...,β_{p}}e_{α_{1},...,α_{p}}+  (4)

+ ∑_{1⩽t⩽p−1} Λ_{t,t+1}e_{β_{1},...,β_{t},α_{t+1},...,α_{p}}e_{α_{1},...,α_{t},β_{t+1},...,β_{p}},

Here α_{1} ≠ β_{1}, …, α_{p} ≠ β_{p} and

Λ = \frac{e^{−2πinτ} θ(y_{1,β_{1}} − y_{1,α_{1}})θ(y_{p,β_{p}} − y_{p,α_{p}} + nτ)}{θ(y_{1,β_{1}} − y_{1,α_{1}} − nτ)θ(y_{p,β_{p}} − y_{p,α_{p}})}  (5)

Λ_{t,t+1} = \frac{e^{−2πinτ} θ(nτ)θ(y_{1,β_{1}} − y_{1,α_{1}})}{θ(y_{1,β_{1}} − y_{1,α_{1}} − nτ)} · \frac{θ(y_{t,β_{t}} + y_{t+1,β_{t+1}} − y_{t,α_{t}} − y_{t+1,α_{t+1}})}{θ(y_{t,β_{t}} − y_{t,α_{t}})θ(y_{t+1,β_{t+1}} − y_{t+1,α_{t+1}})}

In the general case, if some indexes are the same, then we have:

e_{μ_{1},...,μ_{ψ−1},μ_{ψ},α_{1},...,α_{φ},γ_{1},γ_{2},...,γ_{q}}e_{μ′_{1},...,μ′_{ψ−1},μ_{ψ},β_{1},...,β_{φ},γ_{1},γ′_{2},...,γ′_{q}} =

Λe_{μ_{1},...,μ_{ψ−1},μ_{ψ},β_{1},...,β_{φ},γ_{1},γ_{2},...,γ_{q}}e_{μ′_{1},...,μ′_{ψ−1},μ_{ψ},α_{1},...,α_{φ},γ_{1},γ′_{2},...,γ′_{q}}+  (6)

∑_{1⩽t<φ} Λ_{t,t+1}e_{μ_{1},...,μ_{ψ−1},μ_{ψ},β_{1},...,β_{t},α_{t+1},...,α_{φ}γ_{1},γ_{2},...,γ_{q}}e_{μ′_{1},...,μ′_{ψ−1},μ_{ψ},α_{1},...,α_{t},β_{t+1},...,β_{φ}γ_{1},γ′_{2},...,γ′_{q}}

Here Λ, Λ_{t,t+1} are defined by (5), α_{1} ≠ β_{1}, …, α_{φ} ≠ β_{φ}, ψ + φ + q = p. In the case ψ = q = 0 we have the previous relations (4).

We remark that if τ = 0 and p = 1, or p = 2, then the algebra A_{m_{1},...,m_{p}}(E, 0) is the polynomial ring in variables {e_{α_{1},...,α_{p}}} over the field of meromorphic functions in variables {y_{α,β}}. In the case τ = 0, p > 2 the algebra A_{m_{1},...,m_{p}}(E, 0) is commutative but it is not a polynomial ring, because the relations (6) take the form:

e_{μ_{1},...,μ_{ψ−1},μ_{ψ},α_{1},...,α_{φ},γ_{1},γ_{2},...,γ_{q}}e_{μ′_{1},...,μ′_{ψ−1},μ_{ψ},β_{1},...,β_{φ},γ_{1},γ′_{2},...,γ′_{q}} =  (7)
e_{μ_{1},...,μ_{ψ−1},μ_{ψ},β_{1},...,β_{φ},γ_{1},γ_{2},...,γ_{q}}e_{μ′_{1},...,μ′_{ψ−1},μ_{ψ},α_{1},...,α_{φ},γ_{1},γ′_{2},...,γ′_{q}}

The algebra A_{m_{1},...,m_{p}}(E, τ) is a flat deformation of the algebra of functions on the manifold defined by the equalities (7). It is easy to see that this manifold is rational and the general solution of the equalities (7) has a form: e_{α_{1},...,α_{p}} = e_{α_{1},α_{2}}^{(1,2)}e_{α_{2},α_{3}}^{(2,3)} … e_{α_{p−1},α_{p}}^{(p−1,p)}, where {e_{α_{φ},α_{φ+1}}^{(φ,φ+1)}} are independent variables.

Proposition 2. There is the homomorphism of the algebras

x : Q_{n,k}(E, τ) → A_{m_{1},...,m_{p}}(E, τ)

that acts on the generators of the algebra Q_{n,k}(E, τ) in the following way: f(z_{1}, …, z_{p}) ∈ Θ_{(n_{1},...,n_{p})}(Γ) is sent to

x(f) = ∑_{\begin{matrix}1⩽α_{1}⩽m_{1} \\ ... & ... & ... & ... & ... & ... \\ 1⩽α_{p}⩽m_{p} \end{matrix}} f(y_{1,α_{1}}, …, y_{p,α_{p}})e_{α_{1},...,α_{p}}

§3. Functional realization of the Poisson algebra q_{n,k} ˆ⊗q_{m,l}(E)

Let \frac{n}{k} = n_{1} −_{\frac{1}{n_{2}−···− \frac{1}{np}}} , \frac{m}{l} = m_{1} −_{\frac{1}{m_{2}−···− \frac{1}{mq}}} , where n_{1}, …, n_{p}, m_{1}, …, m_{q} ⩾ 2.

We denote q_{n,k} ˆ⊗q_{m,l}(E) = ⨁_{α,β⩾0} P_{α,β}, where P_{α,β} is a space of meromorphic func-

tions f(x_{1,1}, …, x_{p,1}; …; x_{1,α}, …, x_{p,α}; y_{1,1}, …, y_{q,1}; …; y_{1,β}, …, y_{q,β}; z) in variables {x_{i,j}; 1 ⩽ i ⩽ p, 1 ⩽ j ⩽ α}, {y_{i,j}; 1 ⩽ i ⩽ q, 1 ⩽ j ⩽ β} and z with the following properties:

1. Symmetry. For each σ ∈ S_{α}, δ ∈ S_{β}, f is invariant with respect to permutations x_{i,j} ↦ x_{i,σ_{j}}, y_{i,j} ↦ y_{i,δ_{j}}.

2. Periodicity and quasiperiodicity.

f(x_{1,1}, …, x_{t,μ} + 1, …, x_{p,α}; y_{1,1}, …, y_{q,β}; z) = f(x_{1,1}, …, y_{q,β}; z)

f(x_{1,1}, …, x_{p,α}; y_{1,1}, …, y_{t,μ} + 1, …, y_{q,β}; z) = f(x_{1,1}, …, y_{q,β}; z)

f(x_{1,1}, …, x_{t,μ}+η, …, x_{p,α}; y_{1,1}, …, y_{q,β}; z) = e^{−2πi(n_{t}x_{t,μ}−x_{t−1,μ}−x_{t+1,μ})}f(x_{1,1}, …, y_{q,β}, z)

f(x_{1,1}, …, x_{p,α}; y_{1,1}, …, y_{t,μ}+η, …, y_{q,β}; z) = e^{−2πi(m_{t}y_{t,μ}−y_{t−1,μ}−y_{t+1,μ})}f(x_{1,1}, …, y_{q,β}; z)

Here x_{0,μ} = x_{p+1,μ} = y_{0,μ} = y_{q+1,μ} = 0.

3. f(x_{1,1}, …, y_{q,β}; z) as a function in variables {x_{t,μ}, y_{t′,μ′}} is holomorphic outside the divisors {y_{1,μ} − x_{p,μ′} − z = 0; 1 ⩽ μ ⩽ β, 1 ⩽ μ′ ⩽ α} and has a pole

of order ⩽ 1 on these divisors. So f̂(x_{1,1}, …, y_{q,β}; z) = ( ∏_{\begin{matrix}1⩽μ⩽β \\ 1⩽μ′⩽α \end{matrix}} θ(y_{1,μ} − x_{p,μ′} −

z))f(x_{1,1}, …, y_{q,β}; z) as a function in variables {x_{t,μ}, y_{t′,μ′}} is holomorphic.

4. Let 1 ⩽ μ_{1} ≠ μ_{2}, μ_{3} ⩽ β, 1 ⩽ ν_{1}, ν_{2} ≠ ν_{3} ⩽ α. Then f̂(x_{1,1}, …, y_{q,β}; z) = 0 on the affine subspaces of codimension 2 defined by the following relations: y_{1,μ_{1}} = y_{1,μ_{2}} = x_{p,ν_{1}} + z or y_{1,μ_{3}} = x_{p,ν_{2}} + z = x_{p,ν_{3}} + z.

Particularly, the space M = P_{0,0} is a field of meromorphic functions f(z). It is possible to check, that the dimensions dim_{M}P_{α,β} are finite for each α, β and the Hilbert function is

∑_{α,β⩾0} dim_{M}P_{α,β}t_{1}^{α}t_{2}^{β} = (1−t_{1})^{−d(n_{1},...,n_{p})}(1−t_{2})^{−d(m_{1},...,m_{q})}(1−t_{1}t_{2})^{−d(n_{1},...,n_{p}+m_{1},...,m_{q})}

We define the commutative associative product on the space q_{n,k} ˆ⊗q_{m,l}(E) in the following way: if f ∈ P_{α,β}, g ∈ P_{α′,β′} then for fg ∈ P_{α+α′,β+β′} we have:

fg(x_{1,1}, …, x_{p,α+α′}; y_{1,1}, …, y_{q,β+β′}; z) = \frac{1}{α!α′!β!β′!} ∑_{\begin{matrix}σ∈S_{α+α′} \\ δ∈S_{β+β′} \end{matrix}}

f(x_{1,σ_{1}}, …, x_{p,σ_{α}}; y_{1,δ_{1}}, …, y_{q,δ_{β}}; z)g(x_{1,σ_{α+1}}, …, x_{p,σ_{α+α′}}; y_{1,δ_{β+1}}, …, y_{q,δ_{β+β′}}; z)

We define the Poisson bracket {, } on the space q_{n,k} ˆ⊗q_{m,l}(E) in the following way: if f ∈ P_{α,0}, g ∈ P_{β,0} and f, g do not depend on z, then the Poisson bracket {f, g} is given by the formula (3). Similar formula gives the Poisson bracket {f, g} in the case f ∈ P_{0,α}, g ∈ P_{0,β} and f, g do not depend on z. Let f ∈ P_{1,0}, g ∈ P_{0,1} and f, g do not depend on z. In this case, for {f, g} we have by definition:

{f, g}(x_{1}, …, x_{p}; y_{1}, …, y_{q}; z) =

f(x_{1}, …, x_{p}) ∑_{1⩽t⩽q} \frac{d(m_{t+1}, …, m_{q})}{d(m_{1}, …, m_{q})}g′_{y_{t}}(y_{1}, …, y_{q})−

−g(y_{1}, …, y_{q}) ∑_{1⩽t⩽p} \frac{d(n_{1}, …, n_{t−1})}{d(n_{1}, …, n_{p})}f′_{x_{t}}(x_{1}, …, x_{p})−

−(\frac{θ′(y_{1} − x_{p} − z)}{θ(y_{1} − x_{p} − z)} − πi)f(x_{1}, …, x_{p})g(y_{1}, …, y_{q})

For f ∈ P_{1,0} and g ∈ P_{0,1} we assume:

\{f, z\} = (\frac{d(m_{2}, …, m_{q})}{d(m_{1}, …, m_{q})} + \frac{d(n_{1}, …, n_{p−1}) + 1}{d(n_{1}, …, n_{p})})f

\{g, z\} = −(\frac{1 + d(m_{2}, …, m_{q})}{d(m_{1}, …, m_{q})} + \frac{d(n_{1}, …, n_{p−1})}{d(n_{1}, …, n_{p})})g

It is clear that using the Leibniz rule one can extend the Poisson bracket {, } on the whole space q_{n,k} ˆ⊗q_{m,l}(E).

For description of this bracket let us consider the commutative associative algebra generated by all meromorphic functions in variables {x_{α,β}^{(γ)}, y_{α′,β′}^{(γ′)}, z; 1 ⩽ α ⩽ p, 1 ⩽

α′ ⩽ q; β, β′, γ, γ′ ∈ ℕ} and the elements {e(x_{1,β}^{(γ)}, …, x_{p,β}^{(γ)}), e′(y_{1,β′}^{(γ′)}, …, y_{q,β′}^{(γ′)}); β, β′, γ, γ′ ∈ ℕ}. We assume that if u_{α} = v_{α} for some α, then e(u_{1}, …, u_{p})e(v_{1}, …, v_{p}) = 0, e′(u_{1}, …, u_{q})e′(v_{1}, …, v_{q}) = 0.

We define the Poisson structure on this algebra in the following way: {x_{α,β}^{(γ)}, x_{α′,β′}^{(γ′)}} = {x_{α,β}^{(γ)}, y_{α′,β′}^{(γ′)}} = {y_{α,β}^{(γ)}, y_{α′,β′}^{(γ′)}} = {x_{α,β}^{(γ)}, z} = {y_{α′,β′}^{(γ′)}, z} = 0

\{e(u_{1}, …, u_{p}), e(v_{1}, …, v_{p})\} = (\frac{θ′(v_{1} − u_{1})}{θ(v_{1} − u_{1})}+\frac{θ′(v_{p} − u_{p})}{θ(v_{p} − u_{p})}−2πi)e(u_{1}, …, u_{p})e(v_{1}, …, v_{p})+

θ′(0) ∑_{1⩽α<p} \frac{θ(v_{α} + v_{α+1} − u_{α} − u_{α+1})}{θ(v_{α} − u_{α})θ(v_{α+1} − u_{α+1})}e(v_{1}, …, v_{α}, u_{α+1}, …, u_{p})e(u_{1}, …, u_{α}, v_{α+1}, …, v_{p})

\{e′(u_{1}, …, u_{q}), e′(v_{1}, …, v_{q})\} = (\frac{θ′(v_{1} − u_{1})}{θ(v_{1} − u_{1})}+\frac{θ′(v_{q} − u_{q})}{θ(v_{q} − u_{q})}−2πi)e′(u_{1}, …, u_{q})e′(v_{1}, …, v_{q})+

θ′(0) ∑_{1⩽α<q} \frac{θ(v_{α} + v_{α+1} − u_{α} − u_{α+1})}{θ(v_{α} − u_{α})θ(v_{α+1} − u_{α+1})}e′(v_{1}, …, v_{α}, u_{α+1}, …, u_{q})e′(u_{1}, …, u_{α}, v_{α+1}, …, v_{q})

\{e(u_{1}, …, u_{p}), e′(v_{1}, …, v_{q})\} = (\frac{θ′(u_{p} − v_{1} + z)}{θ(u_{p} − v_{1} + z)} − πi)e(u_{1}, …, u_{p})e′(v_{1}, …, v_{q})

\{e(u_{1}, …, u_{p}), x_{α,β}^{(γ)}\} = −\frac{d(n_{1}, …, n_{α−1}) + d(n_{α+1}, …, n_{p})}{d(n_{1}, …, n_{p})}e(u_{1}, …, u_{p})

\{e(u_{1}, …, u_{p}), y_{α,β}^{(γ)}\} = \frac{d(m_{α+1}, …, m_{q})}{d(m_{1}, …, m_{q})}e(u_{1}, …, u_{p})

\{e(u_{1}, …, u_{p}), z\} = (\frac{d(m_{2}, …, m_{q})}{d(m_{1}, …, m_{q})} + \frac{d(n_{1}, …, n_{p−1}) + 1}{d(n_{1}, …, n_{p})})e(u_{1}, …, u_{p})

\{e′(u_{1}, …, u_{q}), y_{α,β}^{(γ)}\} = −\frac{d(m_{1}, …, m_{α−1}) + d(m_{α+1}, …, m_{q})}{d(m_{1}, …, m_{q})}e′(u_{1}, …, u_{q})

\{e′(u_{1}, …, u_{q}), x_{α,β}^{(γ)}\} = \frac{d(n_{1}, …, n_{α−1})}{d(n_{1}, …, n_{p})}e′(u_{1}, …, u_{q})

\{e′(u_{1}, …, u_{q}), z\} = −(\frac{1 + d(m_{2}, …, m_{q})}{d(m_{1}, …, m_{q})} + \frac{d(n_{1}, …, n_{p−1})}{d(n_{1}, …, n_{p})})e′(u_{1}, …, u_{q})

For f ∈ P_{α,β} we define the element X_{f} in the following way:

X_{f} = ∑_{γ_{μ,ν},γ′_{μ′,ν′} ∈ℕ} f(x_{1,1}^{(γ_{1,1})}, …, x_{p,α}^{(γ_{p,α})}; y_{1,1}^{(γ′_{1,1})}, …, y_{q,β}^{(γ′_{q,β})}; z)×

×e(x_{1,1}^{(γ_{1,1})}, …, x_{p,1}^{(γ_{p,1})}) … e(x_{1,α}^{(γ_{1,α})}, …, x_{p,α}^{(γ_{p,α})})e′(y_{1,1}^{(γ′_{1,1})}, …, y_{q,1}^{(γ′_{q,1})}) … e′(y_{1,β}^{(γ′_{1,β})}, …, y_{q,β}^{(γ′_{q,β})})

The Poisson bracket and the product on the space q_{n,k} ˆ⊗q_{m,l}(E) are defined by the following formulas:

X_{{f,g}} = {X_{f}, X_{g}}, X_{fg} = X_{f}X_{g}

§4. Functional realization of the Poisson algebra q_{n_{1},k_{1}} ˆ⊗ … ˆ⊗q_{n_{h},k_{h}}(E)

Let \frac{n_{α}}{k_{α}} = n_{1,α}−_{\frac{1}{n_{2,α}−···− \frac{1}{np_{α},α}}} , where 1 ⩽ α ⩽ h; n_{ν,α} ⩾ 2. Let q_{n_{1},k_{1}} ˆ⊗ … ˆ⊗q_{n_{h},k_{h}}(E) =

⨁_{α_{1},...,α_{h}⩾0} P_{α_{1},...,α_{h}}, where P_{α_{1},...,α_{h}} is the space of meromorphic functions in vari-

ables {x_{μ,λ,t}, z_{ν,ν+1}; 1 ⩽ t ⩽ h, 1 ⩽ μ ⩽ p_{t}, 1 ⩽ λ ⩽ α_{t}, 1 ⩽ ν ⩽ h − 1} with the following properties:

1.Symmetry. Let σ_{t} ∈ S_{α_{t}} for 1 ⩽ t ⩽ h. Then f is invariant with respect to permutations: x_{μ,λ,t} ↦ x_{μ,σ_{t}(λ),t}.

2. Periodicity and quasiperiodicity.

f(x_{1,1,1}, …, x_{μ,λ,t} + 1, …, x_{p_{h},α_{h},h}; z_{1,2}, …, z_{h−1,h}) = f(x_{1,1,1}, …, z_{h−1,h})

f(x_{1,1,1}, …, x_{μ,λ,t} + η, …, x_{p_{h},α_{h},h}; z_{1,2}, …, z_{h−1,h}) =

e^{−2πi(n_{μ,t}x_{μ,λ,t}−x_{μ−1,λ,t}−x_{μ+1,λ,t})}f(x_{1,1,1}, …, z_{h−1,h})

3. f as a function in variables {x_{μ,λ,t}} is holomorphic outside the divisors x_{1,μ,t+1} − x_{p_{t},μ′,t} − z_{t,t+1} = 0 and has a pole of order ⩽ 1 on these divisors. So the

function f̂ = ( ∏_{\begin{matrix}1⩽μ⩽α_{t+1} \\ 1⩽μ′⩽α_{t} \\ 1⩽t<h \end{matrix}} θ(x_{1,μ,t+1} − x_{p_{t},μ′,t} − z_{t,t+1}))f as a function in variables

{x_{μ,λ,t}} is holomorphic.

4. f̂ = 0 on the affine subspaces of codimension 2 defined by the relations x_{1,μ_{1},t+1} = x_{1,μ_{2},t+1} = x_{p_{t},μ_{3},t} + z_{t,t+1} or x_{1,μ_{1},t+1} = x_{p_{t},μ_{2},t} + z_{t,t+1} = x_{p_{t},μ_{3},t} + z_{t,t+1}.

Particularly, the space M = P_{0....,0} is the field of meromorphic functions in variables z_{1,2}, …, z_{h−1,h}. It is possible to check that the dimensions dim_{M}P_{α_{1},...,α_{h}} are finite and the Hilbert function is

∑_{α_{1},...,α_{h}⩾0} dim_{M}P_{α_{1},...,α_{h}}t_{1}^{α_{1}} … t_{h}^{α_{h}} = ∏_{1⩽λ⩽ν⩽h}(1 − t_{λ}t_{λ+1} … t_{ν})^{−d(N_{λ}ΔN_{λ+1}Δ...ΔN_{ν})}

(8)

Here N_{t} = (n_{1,t}, …, n_{p_{t},t}) for 1 ⩽ t ⩽ h. See Notations for definition Δ.

For definition of the product and the Poisson bracket on the space q_{n_{1},k_{1}} ˆ⊗ … ˆ⊗q_{n_{h},k_{h}}(E) we consider the commutative associative algebra generated by all meromorphic functions in variables {x_{μ,λ,t}^{(γ)}, z_{ν,ν+1}; 1 ⩽ μ ⩽ p_{t}, 1 ⩽ t ⩽ h, 1 ⩽ ν ⩽ h − 1, λ, γ ∈ ℕ}

and the elements e_{t}(x_{1,λ,t}^{(γ)}, …, x_{p_{t},λ,t}^{(γ)}). We assume that if u_{α} = v_{α} for some 1 ⩽ α ⩽ p_{t}, then e_{t}(u_{1}, …, u_{p_{t}})e_{t}(v_{1}, …, v_{p_{t}}) = 0. We define the Poisson bracket on this algebra in the following way:

{x_{μ,λ,t}^{(γ)}, x_{μ′,λ′,t′}^{(γ′)}} = {x_{μ,λ,t}^{(γ)}, z_{ν,ν+1}} = {z_{ν,ν+1}, z_{ν′,ν′+1}} = 0

\{e_{t}(u_{1}, …, u_{p_{t}}), x_{μ,λ,t}^{(γ)}\} = −\frac{d(n_{1,t}, …, n_{μ−1,t}) + d(n_{μ+1,t}, …, n_{p_{t},t})}{d(n_{1,t}, …, n_{p_{t},t})}e_{t}(u_{1}, …, u_{p_{t}})

\{e_{t}(u_{1}, …, u_{p_{t}}), x_{μ,λ,t+1}^{(γ)}\} = \frac{d(n_{μ+1,t+1}, …, n_{p_{t+1},t+1})}{d(n_{1,t+1}, …, n_{p_{t+1},t+1})}e_{t}(u_{1}, …, u_{p_{t}})

\{e_{t+1}(u_{1}, …, u_{p_{t+1}}), x_{μ,λ,t}^{(γ)}\} = \frac{d(n_{1,t}, …, n_{μ−1,t})}{d(n_{1,t}, …, n_{p_{t},t})}e_{t+1}(u_{1}, …, u_{p_{t+1}})

{e_{t}(u_{1}, …, u_{p_{t}}), x_{μ,λ,t′}^{(γ)}} = 0, here |t − t′| > 1

{e_{t}(u_{1}, …, u_{p_{t}}), z_{t,t+1}} =

(\frac{d(n_{2,t+1}, …, n_{p_{t+1},t+1})}{d(n_{1,t+1}, …, n_{p_{t+1},t+1})} + \frac{d(n_{1,t}, …, n_{p_{t}−1,t}) + 1}{d(n_{1,t}, …, n_{p_{t},t})})e_{t}(u_{1}, …, u_{p_{t}})

{e_{t+1}(u_{1}, …, u_{p_{t+1}}), z_{t,t+1}} =

−(\frac{d(n_{2,t+1}, …, n_{p_{t+1},t+1}) + 1}{d(n_{1,t+1}, …, n_{p_{t+1},t+1})} + \frac{d(n_{1,t}, …, n_{p_{t}−1,t})}{d(n_{1,t}, …, n_{p_{t},t})})e_{t+1}(u_{1}, …, u_{p_{t+1}})

{e_{t}(u_{1}, …, u_{p_{t}}), z_{t′,t′+1}} = 0, here t ≠ t′, t ≠ t′ + 1

{e_{t}(u_{1}, …, u_{p_{t}}), e_{t}(v_{1}, …, v_{p_{t}})} =

(\frac{θ′(v_{1} − u_{1})}{θ(v_{1} − u_{1})} + \frac{θ′(v_{p_{t}} − u_{p_{t}})}{θ(v_{p_{t}} − u_{p_{t}})} − 2πi)e_{t}(u_{1}, …, u_{p_{t}})e_{t}(v_{1}, …, v_{p_{t}})+

θ′(0) ∑_{1⩽α<p_{t}} \frac{θ(v_{α} + v_{α+1} − u_{α} − u_{α+1})}{θ(v_{α} − u_{α})θ(v_{α+1} − u_{α+1})}e_{t}(v_{1}, …, v_{α}, u_{α+1}, …, u_{p_{t}})e(u_{1}, …, u_{α}, v_{α+1}, …, v_{p_{t}})

{e_{t}(u_{1}, …, u_{p_{t}}), e_{t+1}(v_{1}, …, v_{p_{t+1}})} =

(\frac{θ′(u_{p_{t}} − v_{1} + z_{t,t+1})}{θ(u_{p_{t}} − v_{1} + z_{t,t+1})} − πi)e_{t}(u_{1}, …, u_{p_{t}})e_{t+1}(v_{1}, …, v_{p_{t+1}})

{e_{t}(u_{1}, …, u_{p_{t}}), e_{t′}(v_{1}, …, v_{p_{t′}}) = 0, here |t − t′| > 1

For f ∈ P_{α_{1},...,α_{h}} we define the element X_{f} by the following formula:

X_{f} = ∑_{γ_{μ,λ,ν∈ℕ}} f(x_{1,1,1}^{(γ_{1,1,1})}, …, x_{p_{h},α_{h},h}^{(γ_{p_{h},α_{h},h})}, z_{1,2}, …, z_{h−1,h}) ∏_{\begin{matrix}1⩽t⩽h \\ 1⩽λ⩽α_{t} \end{matrix}} e_{t}(x_{1,λ,t}^{(γ_{1,λ,t})}, …, x_{p_{t},λ,t}^{(γ_{p_{t},λ,t})})

Proposition 3. The following formulas define the product and the Poisson bracket on the space q_{n_{1},k_{1}} ˆ⊗ … ˆ⊗q_{n_{h},k_{h}}(E):

X_{fg} = X_{f}X_{g}, X_{{f,g}} = {X_{f}, X_{g}}

§5. Bosonization of the algebra Q_{n_{1},k_{1}} ˆ⊗ … ˆ⊗Q_{n_{h},k_{h}}(E, τ)

Let A_{l_{1,1},...,l_{p_{1},1};...;l_{1,h},...,l_{p_{h},h}}(E, τ) be the associative algebra generated by all meromorphic functions in variables {y_{λ,μ,γ}, z_{γ′,γ′+1}; 1 ⩽ γ ⩽ h, 1 ⩽ γ′ ⩽ h − 1, 1 ⩽ λ ⩽ p_{γ}, 1 ⩽ μ ⩽ l_{λ,γ}} and the elements {e_{α_{1},...,α_{pγ}}^{(γ)} ; 1 ⩽ γ ⩽ h, 1 ⩽ α_{λ} ⩽ l_{λ,γ}, 1 ⩽ λ ⩽ p_{γ}}. We assume that the following relations hold:

[y_{λ,μ,γ}, y_{λ′,μ′,γ′}] = [y_{λ,μ,γ}, z_{t,t+1}] = [z_{t,t+1}, z_{t′,t′+1}] = 0

e_{α_{1},...,α_{pγ}}^{(γ)} y_{λ,β,γ} = (y_{λ,β,γ} − \frac{d(n_{1,γ}, …, n_{λ−1,γ}) + d(n_{λ+1,γ}, …, n_{p_{γ},γ})}{d(n_{1,γ}, …, n_{p_{γ},γ})}τ)e_{α_{1},...,α_{pγ}}^{(γ)}

Here β ≠ α_{λ}.

e_{α_{1},...,α_{pγ}}^{(γ)} y_{λ,α_{λ},γ} = (y_{λ,α_{λ},γ}+(1−\frac{d(n_{1,γ}, …, n_{λ−1,γ}) + d(n_{λ+1,γ}, …, n_{p_{γ},γ})}{d(n_{1,γ}, …, n_{p_{γ},γ})})τ)e_{α_{1},...,α_{pγ}}^{(γ)}

e_{α_{1},...,α_{pγ}}^{(γ)} y_{λ,β,γ+1} = (y_{λ,β,γ+1} + \frac{d(n_{λ+1,γ+1}, …, n_{p_{γ+1},γ+1})}{d(n_{1,γ+1}, …, n_{p_{γ+1},γ+1})}τ)e_{α_{1},...,α_{pγ}}^{(γ)}

e_{α_{1},...,α_{p_{γ+1}}}^{(γ+1)} y_{λ,β,γ} = (y_{λ,β,γ} + \frac{d(n_{1,γ}, …, n_{λ−1,γ})}{d(n_{1,γ}, …, n_{p_{γ},γ})}τ)e_{α_{1},...,α_{p_{γ+1}}}^{(γ+1)}

e_{α_{1},...,α_{pγ}}^{(γ)} y_{λ,β,γ′} = y_{λ,β,γ′}e_{α_{1},...,α_{pγ}}^{(γ)}

Here |γ − γ′| > 1.

e_{α_{1},...,α_{pγ}}^{(γ)} z_{γ,γ+1} =

(z_{γ,γ+1} + (\frac{d(n_{2,γ+1}, …, n_{p_{γ+1},γ+1})}{d(n_{1,γ+1}, …, n_{p_{γ+1},γ+1})} + \frac{d(n_{1,γ}, …, n_{p_{γ}−1,γ}) + 1}{d(n_{1,γ}, …, n_{p_{γ},γ})})τ)e_{α_{1},...,α_{pγ}}^{(γ)}

e_{α_{1},...,α_{p_{γ+1}}}^{(γ+1)} z_{γ,γ+1} =

(z_{γ,γ+1} − (\frac{d(n_{2,γ+1}, …, n_{p_{γ+1},γ+1}) + 1}{d(n_{1,γ+1}, …, n_{p_{γ+1},γ+1})} + \frac{d(n_{1,γ}, …, n_{p_{γ}−1,γ})}{d(n_{1,γ}, …, n_{p_{γ},γ})})τ)e_{α_{1},...,α_{p_{γ+1}}}^{(γ+1)}

e_{α_{1},...,α_{pγ}}^{(γ)} z_{γ′,γ′+1} = z_{γ′,γ′+1}e_{α_{1},...,α_{pγ}}^{(γ)}

Here γ ≠ γ′ and γ ≠ γ′ + 1.

e_{α_{1},...,α_{pγ}}^{(γ)} e_{β_{1},...,β_{p_{γ′}}}^{(γ′)} = e_{β_{1},...,β_{p_{γ′}}}^{(γ′)} e_{α_{1},...,α_{pγ}}^{(γ)}

Here |γ − γ′| > 1.

e_{α_{1},...,α_{pγ}}^{(γ)} e_{β_{1},...,β_{p_{γ+1}}}^{(γ+1)} =

−e^{2πi(y_{1,β_{1},γ+1}−y_{pγ ,αp_{γ} ,γ}−z_{γ,γ+1})} \frac{θ(y_{p_{γ},α_{pγ} ,γ} − y_{1,β_{1},γ+1} + z_{γ,γ+1} + \frac{1}{2}τ)}{θ(y_{1,β_{1},γ+1} − y_{p_{γ},α_{pγ} ,γ} − z_{γ,γ+1} + \frac{1}{2}τ)}e_{β_{1},...,β_{p_{γ+1}}}^{(γ+1)} e_{α_{1},...,α_{pγ}}^{(γ)}

e_{α_{1},...,α_{pγ}}^{(γ)} e_{β_{1},...,β_{pγ}}^{(γ)} = Φe_{β_{1},...,β_{pγ}}^{(γ)} e_{α_{1},...,α_{pγ}}^{(γ)} +  (9)

+ ∑_{1⩽t⩽p_{γ}−1} Φ_{t,t+1}e_{β_{1},...,β_{t},α_{t+1},...,α_{pγ}}^{(γ)} e_{α_{1},...,α_{t},β_{t+1},...,β_{pγ}}^{(γ)}

Here α_{1} ≠ β_{1}, …, α_{p_{γ}} ≠ β_{p_{γ}} and

Φ = \frac{e^{−2πiτ} θ(y_{1,β_{1},γ} − y_{1,α_{1},γ})θ(y_{p_{γ},β_{pγ} ,γ} − y_{p_{γ},α_{pγ} ,γ} + τ)}{θ(y_{1,β_{1},γ} − y_{1,α_{1},γ} − τ)θ(y_{p_{γ},β_{pγ} ,γ} − y_{p_{γ},α_{pγ} ,γ})}  (10)

Φ_{t,t+1} = \frac{e^{−2πiτ} θ(τ)θ(y_{1,β_{1},γ} − y_{1,α_{1},γ})}{θ(y_{1,β_{1},γ} − y_{1,α_{1},γ} − τ)}·\frac{θ(y_{t,β_{t},γ} + y_{t+1,β_{t+1},γ} − y_{t,α_{t},γ} − y_{t+1,α_{t+1},γ})}{θ(y_{t,β_{t},γ} − y_{t,α_{t},γ})θ(y_{t+1,β_{t+1},γ} − y_{t+1,α_{t+1},γ})}

In the general case, if some indexes are the same, then we have: e_{μ_{1},...,μ_{ψ−1},μ_{ψ},α_{1},...,α_{φ},γ_{1},γ_{2},...,γ_{q}}^{(γ)}e_{μ′_{1},...,μ′_{ψ−1},μ_{ψ},β_{1},...,β_{φ},γ_{1},γ′_{2},...,γ′_{q}}^{(γ)} =

Φe_{μ_{1},...,μ_{ψ−1},μ_{ψ},β_{1},...,β_{φ},γ_{1},γ_{2},...,γ_{q}}^{(γ)}e_{μ′_{1},...,μ′_{ψ−1},μ_{ψ},α_{1},...,α_{φ},γ_{1},γ′_{2},...,γ′_{q}}^{(γ)}+

∑_{1⩽t<φ} Φ_{t,t+1}e_{μ_{1},...,μ_{ψ−1},μ_{ψ},β_{1},...,β_{t},α_{t+1},...,α_{φ}γ_{1},γ_{2},...,γ_{q}}^{(γ)}e_{μ′_{1},...,μ′_{ψ−1},μ_{ψ},α_{1},...,α_{t},β_{t+1},...,β_{φ}γ_{1},γ′_{2},...,γ′_{q}}^{(γ)}

Here Φ, Φ_{t,t+1} are defined by (10), α_{1} ≠ β_{1}, …, α_{φ} ≠ β_{φ}, ψ + φ + q = p_{γ}. In the case ψ = q = 0 we have the previous relations (9).

Proposition 4. There is a family of the associative algebras Q_{n_{1},k_{1}} ˆ⊗ … ˆ⊗Q_{n_{h},k_{h}}(E, τ) that is the quantization of the Poisson algebra q_{n_{1},k_{1}} ˆ⊗ … ˆ⊗q_{n_{h},k_{h}}(E) and the homomorphism of the associative algebras

x : Q_{n_{1},k_{1}} ˆ⊗ … ˆ⊗Q_{n_{h},k_{h}}(E, τ) → P_{l_{1,1},...,l_{p_{h},h}}(E, τ)

such that for the element f ∈ P_{0,...,0,1,0,...,0} (here 1 is on γ-th place) we have:

x(f) = ∑_{\begin{matrix}1⩽α_{1}⩽l_{1,γ} \\ ... & ... & ... & ... & ... & ... \\ 1⩽α_{pγ} ⩽l_{pγ ,γ} \end{matrix}} f(y_{1,α_{1},γ}, …, y_{p_{γ},α_{pγ} ,γ}; z_{1,2}, …, z_{h−1,h})e_{α_{1},...,α_{pγ}}^{(γ)}

The algebras Q_{n_{1},k_{1}} ˆ⊗ … ˆ⊗Q_{n_{h},k_{h}}(E, τ) and q_{n_{1},k_{1}} ˆ⊗ … ˆ⊗q_{n_{h},k_{h}}(E) have the same Hilbert function (see (8)).

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. Philozen. 27, No. 1, 37-45 (1993).

3. 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).

4. A.V.Odesskii and B.L.Feigin, ”Quantized moduli spaces of the bundles on the elliptic curve and their applications.”, MPI Preprint Series 1998 (132), Bonn, Germany. (math.QA/9812059)

5. B.L.Feigin and A.V.Odesskii, Vector Bundles on Elliptic Curve and Sklyanin Algebras, RIMS-1032, September 1995, Kyoto University, Kyoto, Japan. (q-alg/9509021)