ITEP-TH-30/03

August 22, 2003

Bihamiltonian structures and quadratic algebras in hydrodynamics and on non-commutative torus

B.Khesin Max Planck Institute of Mathematics, Bonn, Germany University of Toronto, Toronto, Canada e-mail: khesin@math.toronto.edu A.Levin Max Planck Institute of Mathematics, Bonn, Germany Institute of Oceanology, Moscow, Russia e-mail: alevin@wave.sio.rssi.ru M. Olshanetsky Max Planck Institute of Mathematics, Bonn, Germany Institute of Theoretical and Experimental Physics, Moscow, Russia, e-mail: olshanet@gate.itep.ru

Abstract

We demonstrate the common bihamiltonian nature of several integrable systems. The first one is an elliptic rotator that is an integrable Euler-Arnold top on the complex group GL(N, ℂ) for any N, whose inertia ellipsiod is related to a choice of an elliptic curve. Its bihamiltonian structure is provided by the compatible linear and quadratic Poisson brackets, both of which are governed by the Belavin-Drinfeld classical elliptic r-matrix. We also generalize this bihamiltonian construction of integrable Euler-Arnold tops to several infinitedimensional groups, appearing as certain large N limits of GL(N, ℂ). These are the group of a non-commutative torus (NCT) and the group of symplectomorphisms SDiff(T^{2}) of the two-dimensional torus. The elliptic rotator on symplectomorphisms gives an elliptic version of an ideal 2D hydrodynamics, which turns out to be an integrable system. In particular, we define the quadratic Poisson algebra on the space of Hamiltonians on T^{2} depending on two irrational numbers. In conclusion, we quantize the infinite-dimensional quadratic Poisson algebra in a fashion similar to the corresponding finite-dimensional case.

Contents

1 Introduction 2

2 Main results 3

3 GL(N, ℂ)-Elliptic Rotators 6 3.1 Lax representation and integrals of motion . . . . . . . . . . . . . . . . . . . . . . 6 3.2 The classical GL(N, ℂ) Feigin-Odesski algebras . . . . . . . . . . . . . . . . . . . 10 3.3 The bihamiltonian structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 4 Elliptic Rotators on non-commutative torus 13

4.1 The Lax equation and integrals of motion . . . . . . . . . . . . . . . . . . . . . . 13 4.2 Quadratic Poisson algebras and the bihamiltonian structure on NCT . . . . . . . 16

5 Elliptic Rotators on SDiff(T^{2}) 17 5.1 Description of the hierarchy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 5.2 Quadratic Poisson algebras, and the bihamiltonian structure on SDiff(T^{2}) … 19 5.3 Rational limit of the elliptic hydrodynamics . . . . . . . . . . . . . . . . . . . . . 20

6 Quantum counterparts 22

Acknowledgments 25

7 Appendix 25 7.1 Appendix A. Elliptic functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 7.2 Appendix B. Elliptic constants related to gl(N, ℂ). . . . . . . . . . . . . . . . . . 27 7.3 Appendix C. Non-commutative torus . . . . . . . . . . . . . . . . . . . . . . . . . 28

1 Introduction

One of the most interesting problems in the theory of integrable systems is a description of infinite-dimensional integrable systems with two or more space variables. In this paper we consider integrable hierarchies in (2+1) dimension, which we call elliptic and modified hydrodynamics. These are close relatives of the ideal hydrodynamics on a two-dimensional torus, which is known to be a highly non-integrable system. For instance, even the 4-vortex approximation of the ideal fluid dynamics is non-integrable [1]. We obtain these integrable versions of hydrodynamics starting from finite-dimensional integrable systems whose dimension of the phase space goes to infinity. Our approach is somewhat similar to the derivation of Toda field theory from the open Toda chain. In contrast with the Toda theory, however, we obtain non-local systems.

The classical Euler (or, rather, Helmholtz) equation for the motion of an ideal fluid on the standard two-torus is

∂_{t}S = {S, Δ^{−1}S} ,

where S is the vorticity function of the fluid flow, { } is the Poisson bracket, and Δ is the Laplace operator on T^{2}. The modified and elliptic hydrodynamics are defined on an elliptic curve, i.e., on a torus with a complex structure fixed. The corresponding equations are, respectively,

∂_{t}S = {S, ¯∂^{−2}S}

and

∂_{t}S = {S, ℘(¯∂)S} ,

where ¯∂ is the corresponding operator of the complex structure, and ℘ is the Weierstrass ℘- function. (We postpone the precise description of ℘(¯∂) till Section 5.) Note that to define the Laplace operator one needs to choose a metric, while for the operator ¯∂ in the modified hydrodynamics is defined by a complex structure on T^{2}.

Such a modification of the fluid inertia operator from the Laplace operator Δ = ∂ ¯∂, which depends on a metric on T^{2}, to ¯∂^{2} or ℘(¯∂)^{−1}, both of which depend on a complex structure on T^{2}, brings in the integrability and even the bihamiltonian structure for the systems. We construct an infinite set of involutive integrals of motion with respect to two Poisson brackets.

One of the brackets is the standard linear Lie-Poisson brackets on the dual space to the algebra of the divergence-free vector fields on T^{2}. The other Poisson structure is a quadratic Poisson algebra on Hamiltonians of vector fields. These two brackets are compatible and governed by the same classical r-matrix. Furthermore, we describe a recursion procedure for constructing the sequence of Hamiltonians for this linear-quadratic bihamiltonian structure of the hierarchy of the elliptic hydrodynamics, which thereby exhibits “the strongest form of integrability.”

We come to this construction through the non-commutative deformation of T^{2} to the noncommutative torus (NCT). The non-commutative deformation of the Lie algebra of vector fields is isomorphic to the Lie algebra of NCT. The latter is a special large N limit of gl(N, ℂ). We start with the elliptic rotators on GL(N, ℂ) [2, 3] and develop their bihamiltonian structure based on the Belavin-Drinfeld classical r-matrix [4]. The corresponding quadratic Poisson algebra is the classical limit of the Feigin-Odesski algebra [5]. We describe natural extensions of the elliptic rotators to the infinite-dimensional groups, which preserve their main properties. This brings us to bihamiltonian systems and integrable hierarchies on the NCT, which are of interest by themselves.

As a byproduct, we also describe a quantization of the quadratic Poisson algebra on NCT. In this way we obtain an infinite-dimensional associative algebra with quadratic relations depending on the complex structure on T^{2} and the Planck constant ℏ ∈ T^{2}.

Finally, the commutative (”dispersionless”) limit of the elliptic rotators on NCT leads to the desirable bihamiltonian hierarchies of the elliptic and modified hydrodynamics.

2 Main results

1. The general setup [6, 7] Let G be a Lie group and 𝔤 its Lie algebra. Consider an invertible linear operator J that maps the coalgebra 𝔤^{∗} to 𝔤. Its inverse operator J^{−1} is called the inertia tensor. The Euler-Arnold top corresponding to the group G is the Hamiltonian system on 𝔤^{∗} with respect to the linear Lie-Poisson brackets on 𝔤^{∗} and the Hamiltonian function given by the quadratic form

H = −\frac{1}{2}⟨S, J(S)⟩ , S ∈ 𝔤^{∗} ,

where ⟨ , ⟩ stands for the pairing between 𝔤 and 𝔤^{∗}. Namely, the variation ∇H can be regarded as an element of 𝔤 and the corresponding Hamiltonian equation of motion is as follows:

∂_{t}S = {H, S} := ad_{∇H}^{∗}S .

Recall that the Lie-Poisson brackets are degenerate on 𝔤^{∗} and their symplectic leaves are coadjoint orbits of G. To descend to a particular coadjoint orbit O one should fix the values of Casimirs for the linear bracket.

For some special choices of J the system becomes completely integrable. Some of the examples are Manakov’s tops on SO(N) [8], their limit N → ∞ found by Ward [9], the Korteweg–de Vries equation [10, 11], as well as the Camassa–Holm and Hunter–Saxton equations on the Virasoro group [12]. (In the infinite-dimensional case, the invertibility of J is understood as KerJ = 0.)

2. Elliptic Rotators on GL(N, ℂ) The elliptic rotator (ER) on GL(N, ℂ) is an integrable Euler-Arnold top on this group, whose inertia operator is constructed with the help of the Weierstrass ℘-function of an auxilliary elliptic curve, see Ref. [2, 3, 13]. Namely, let T_{α}, α = (α_{1}, α_{2}) be the basis of sl(N, ℂ) (B.4,) α_{j} = (0, 1, …, N − 1), α ≠ (0, 0). The structure constants in this basis are

C_{θ}(α, β) = \frac{1}{πθ} sin πθ(α × β) , θ = k/N ,

where 1 ≤ k < N and k, N are coprime, see (B.8). For S = ∑_{α} S_{−α}T_{α} ∈ sl(N, ℂ)^{∗} the linear Poisson brackets assume the form

{S_{α}, S_{β}} = C_{θ}(α, β)S_{α+β} .  (2.1)

Let ℘(x; τ) be the Weierstrass function on the elliptic curve E_{τ} = ℂ/(ℤ+τℤ). Consider its values ℘_{θ}(α) = ℘((α_{1} +α_{2}τ)θ; τ) on the lattice parametrized by α, the labels of the sl(N, ℂ)-basis. The inverse inertia operator is defined as

J : S_{α} → J_{α}S_{α} , J_{α} = ℘_{θ}(α) ,  (2.2)

It was proved in Ref .[2] that the equations of motion

∂_{t}S_{α} = ∑_{γ} S_{α−γ}℘_{θ}(γ)S_{γ} ,  (2.3)

defined by the Hamiltonian H = −\frac{1}{2}⟨S, J(S)⟩ = 2π^{2}θ^{2}tr(S · J(S)) and the brackets (2.1), have the Lax representation with the Lax operator L^{rot}(S, z) ∈ sl(N, ℂ) depending on the spectral parameter z ∈ E_{τ}. The traces tr(L^{rot}(S, z))^{k}, k = 2, …, N being expanded in the basis of elliptic functions on E_{τ} produce the involutive integrals of motion and the hamiltonian H is among them.

We call the commuting flows defined by these integrals the ER hierarchy.

Let r(z), z ∈ E_{τ} be the Belavin-Drinfeld classical elliptic r-matrix, see [13, 4] and Section 3 below. The main result, which we will also generalize to the infinite-dimensional situation, can be formulated as follows.

Theorem 2.1 i) In terms of the r-matrix, the linear brackets (2.1) can be written in the form

{L_{1}^{rot}(z), L_{2}^{rot}(w)}_{1} = [r(z − w), L_{1}^{rot}(z) + L_{2}^{rot}(w)] .  (2.4)

ii) Consider the phase space extended by a new variable S_{0}, which has zero bracket with all the rest. The same r-matrix defines the quadratic Poisson algebra P_{N,θ,τ} related to GL(N, ℂ) (3.19), (3.20):

{L_{1}(z), L_{2}(w)}_{2} = [r(z − w), L_{1}(z) ⊗ L_{2}(w)] , where L(z) = S_{0}Id + L^{rot}(z) .  (2.5)

iii) The above two brackets are compatible, i.e. any linear combination of them is a Poisson bracket. iv) There exists a sequence of integrals of motion in involution with respect to each of the two brackets {h_{j}, h_{k}}_{1,2} = 0 (here the lower indices refer to the linear and the quadratic brackets respectively). They provide the bihamiltonian structure of the elliptic rotator (ER) hierarchy

{h_{j+1}, S}_{2} = −{h_{j}, S}_{1} .

The first two statements of Theorem for GL(N, ℂ) are well known [5, 14]. Apparently, the bihamiltonian structure of the GL_{N} elliptic rotators is new, and it gives the following Corollary 2.1 The Casimirs with respect to one of the brackets generate non-trivial dynamics with respect to the other.

In particular, the functional S_{0}, being the Casimir element of the linear brackets (2.1), leads to the equations

∂_{t}S = {S_{0}, S}_{2} ,  (2.6)

that coincide with (2.3).

It turns out that Theorem 2.1 also holds in the infinite-dimensional situation presented below.

3. Elliptic Rotators on the Non-commutative Torus Consider the sin-algebra sin_{θ}, called also the Lie algebra of the non-commutative torus (NCT). The Poisson brackets on the Lie coalgebra sin_{θ}^{∗} of the NCT has the form (2.1) with irrational number 0 ≤ θ < 1 and the basis T_{α} = const · exp(2πiα_{1}x_{1}) ∗ exp(2πiα_{2}x_{2}) is parameterized by the infinite lattice α ∈ ℤ ⊕ ℤ. Here exp(2πix_{1}), exp(2πix_{2}) are the generators of the NCT and ∗ is the Moyal multiplication (see Appendix C). On the NCT we introduce the complex structure ¯∂_{ϵ,τ} depending on τ (Im τ > 0), and two real numbers ϵ = (ϵ_{1}, ϵ_{2}), such that θϵ_{a} are irrational and 0 < θϵ_{a} ≤ 1 (see (4.3)). The inverse inertia operator J is the pseudo-differential operator

J(S)(x) = ℘(θ ¯∂_{ϵ,τ})S(x) ,

where now S(x) = ∑_{α} S_{−α}T_{α}(x). The equation of motion in the form of the Moyal brackets has the form

∂_{t}S = {S, ℘(θ ¯∂_{ϵ,τ})S}^{θ} ,  (2.7)

where { }^{θ} is the Lie–Poisson bracket on sin_{θ}^{∗}. In Proposition 3.1 we find the Lax form of (2.7). We construct the classical r-matrix (4.11) and prove the counterpart of the Theorem. The quadratic Poisson algebra P_{θ,ϵ,τ} (4.19), (4.20) on the NCT gives rise to the infinite bihamiltonian hierarchy, where the equations (2.7) correspond to the quadratic Hamiltonian functional with respect to the linear brackets. At the same time the equations (2.7) can be interpreted as the Hamiltonian equations (2.6) in the quadratic Poisson algebra P_{θ,ϵ,τ} with a linear Hamiltonian. For SL(2) this representation was found in [15] and for general case in [18].

4. Elliptic Rotators on SDiff(T^{2}) In the limit θ → 0 the Lie algebra of the NCT becomes isomorphic to the Poisson algebra A of smooth functions on T^{2} modulo constants Ham(T^{2}) ∼ C^{∞}(T^{2})/ℂ. The algebra of Hamiltonians can be also described by the corresponding Hamiltonian (or, divergence-free) vector fields on the torus. More precisely, the Lie algebra SV ect(T^{2}) of divergence-free vector fields on T^{2} is the (universal) cocentral extension of Ham(T^{2}), defined by the exact sequence

0 → Ham(T^{2}) → SV ect(T^{2}) → ℂ^{2} → 0 ,

where the image of SV ect(T^{2}) in ℂ^{2} is generated by the two fluxes ϵ_{1}∂_{1}, ϵ_{2}∂_{2}. For ψ ∈ Ham(T^{2}) we have

V_{1}(ψ) = −\frac{1}{4π^{2}} ∂_{2}ψ , V_{2}(ψ) = \frac{1}{4π^{2}} ∂_{1}ψ ,

and V_{{ψ,ψ′}} = [V (ψ), V (ψ)′]. We construct the elliptic rotator on the Lie group SDiff(T^{2}) of area-preserving diffeomorphisms, corresponding to the Lie algebra SV ect(T^{2}).

Consider the dual space of linear functionals Ham(T^{2})^{∗} in the Fourier basis

{e(α · x) := exp 2πi(α · x) | (x = (x_{1}, x_{2}) , (α · x) = α_{1}x_{1} + α_{2}x_{2} , α_{j} ∈ ℤ} ,

S = ∑_{α} S_{α}e(−(α · x)) ∈ Ham(T^{2})^{∗} .

The Poisson structure on A^{∗} assumes the form (cf. (2.1))

{S_{α}, S_{β}}_{1} = (α × β)S_{α+β} .

This Poisson structure is degenerate and has an infinite set of Casimirs

C_{k} = ∫_{T^{2}} S^{k} , where k = 2, 3, ….

On a coadjoint orbit O ⊂ Ham(T^{2})^{∗} of SDiff(T^{2}) the brackets are non-degenerate and values of the Casimirs are fixed.

Define the operator J (after the rescaling) as the pseudo-differential operator ℘(¯∂_{ϵ,τ}), such that the Hamiltonian has the form

H = −\frac{1}{2} ∫_{T^{2}} S · ℘(¯∂_{ϵ,τ})S .  (2.8)

We prove that the corresponding Hamiltonian system admits an infinite set of commuting integrals, see Section 5.

Recall that these elliptic rotators are parameterized, in particular, by the auxiliary elliptic curve. Consider the simplest version of the ER, corresponding to the rational degeneration of the elliptic curve E_{τ}. In the limit the (inverse) inertia operator becomes J = ¯∂_{τ}^{−2} and the limiting Hamiltonian (2.8) takes the form

H = −\frac{1}{2} ∫_{T^{2}} S · ¯∂_{τ}^{−2}S ,

see (5.23) and [15].

It is interesting to compare this with the Euler equation for an ideal fluid on a torus, where the inertia operator is the Laplacian (i.e., J = Δ^{−1} = (¯∂∂)^{−1}), and the Hamiltonian is

H = −\frac{1}{2} ∫_{T^{2}} S · Δ^{−1}S ,

where S plays the role of the vorticity [7]. While the Euler hydrodynamics equation is highly non-integrable, the modification of the (inverse) inertia operator from ¯∂∂ to ¯∂_{τ}^{2} leads to an integrable hierarchy. We call the rational limit the modified hydrodynamics on T^{2}. (In a sense, these systems are similar in spirit to the Etingof–Frenkel current algebras [16] on T^{2}, which make use of the complex structure on the elliptic curve to construct a central extension of smooth currents on the torus.)

Finally, we prove the Theorem 2.1 for the generic systems of the elliptic hydrodynamics (2.8). In particular, the quadratic Poisson algebra P_{0,τ} (5.27), (5.28) on T^{2}, along with the linear Poisson bracket, provide the bihamiltonian structure of the hierarchies of the elliptic and modified hydrodynamics.

3 GL(N, ℂ)-Elliptic Rotators

3.1 Lax representation and integrals of motion

1. The system description The elliptic GL(N, ℂ)-rotator is defined on the dual spaces of gl(N, ℂ) and sl(N, ℂ). Let S = ∑_{α} S_{−α}T_{α}, where T_{α} is the basis of sl(N, ℂ), see (B.4). Then the Poisson structure on the dual space sl(N, ℂ)^{∗} is given by the linear Lie-Poisson brackets (B.8)

{S_{α}, S_{β}}_{1} = C_{θ}(α, β)S_{α+β} ,  (3.1)

where C_{θ}(α, β) are the structure constants of sl(N, ℂ) (B.8). The Hamiltonian has the form

H^{rot} = 2π^{2}θ^{2}tr(S · J(S)) ≡ −\frac{1}{2} ∑_{γ} S_{γ}℘_{γ}S_{−γ} .  (3.2)

It defines the equations of motion

∂_{t}S = [J(S), S].  (3.3)

Let J : S_{α} → J_{α}S_{α} , be the (inverse) inertia operator with J_{α} = ℘_{θ}(α), where ℘_{θ}(α) are the values of the Weierstrass ℘-function and defined by (B.11). Then (3.3) assumes the form

∂_{t}S_{α} = ∑_{γ} S_{α−γ}S_{γ}℘_{θ}(γ)C_{θ}(γ, α) .  (3.4)

2. The linear brackets and r-matrix

Proposition 3.1 (cf.[2]) The equations of motion (3.2) have the Lax form

∂_{t}L^{rot}(z) = [L^{rot}(z), M(z)] ,  (3.5)

where

L^{rot} = − ∑_{α} S_{α}φ_{θ}(α, z)T_{α} ,  (3.6)
M = − ∑_{α} S_{α}f_{θ}(α, z)T_{α} ,  (3.7)

and the functions φ_{θ}, f_{θ} are given by (B.13), (B.14).

Proof. Substituting (3.6) and (3.7) in the Lax equation we obtain

∂_{t}S_{α}φ_{θ}(α, z) = ∑_{γ} S_{α−γ}S_{γ}(φ_{θ}(α − γ, z)f_{θ}(γ, z) − φ_{θ}(γ, z)f_{θ}(α − γ, z)) .

Now, using the explicit expressions (B.13) and (B.14) for φ_{θ} and f_{θ} respectively, as well as the Calogero functional equation (A.22), we come to

∂_{t}S_{α} = ∑_{γ} S_{α−γ}S_{γ}φ_{θ}(α, z)(℘_{θ}(α − γ) − ℘_{θ}(γ)),

which coincides with (3.4). □

The Lie-Poisson brackets (3.1) admit the following r-matrix description [4, 13, 14, 17]. Define the classical r-matrix by

r_{N,θ,τ} (z − w) = ∑_{γ} φ_{θ}(γ, z − w)T_{γ} ⊗ T_{−γ} ,  (3.8)

where φ_{θ} is defined by (B.13).

Lemma 3.1 [14]. The r-matrix (3.8) satisfies the classical Yang-Baxter equation

[r_{N,θ,τ} (z − w), r_{N,θ,τ} (z)] + [r_{N,θ,τ} (z − w), r_{N,θ,τ} (w)]
+[r_{N,θ,τ} (z), r_{N,θ,τ} (w)] = 0 .  (3.9)

Proof. We reduce the YB equation (3.9) to the following functional equation

φ_{θ}(γ, z − w)φ_{θ}(α, z) − φ_{θ}(α + γ, z − w)φ_{θ}(α, w) + φ_{θ}(α + γ, z)φ_{θ}(−γ, w) = 0 ,

by using the commutation relations in sl(N, ℂ) (B.7). It easy to see that it can be rewritten for the functions ϕ_{θ} (B.12), since all three terms have the common exponent e_{θ}((γ_{2} + α_{2})z − γ_{2}w). Now, it coincides with the Fay identity (A.21), where we put

u_{1} = (γ + α)θ , u_{2} = −γθ , z_{1} = z , z_{2} = w .

□

Proposition 3.2 In terms of the Lax operator (3.7) the brackets (3.1) are equivalent to the following relation for the Lax operator

{L_{1}^{rot}(z), L_{2}^{rot}(w)}_{1} = [r_{N,θ,τ} (z − w), L^{rot}(z) ⊗ Id + Id ⊗ L^{rot}(w)] ,  (3.10)
L_{1} = L ⊗ Id , L_{2} = Id ⊗ L .

Proof. To prove (3.10) we rewrite it in the basis T_{α} ⊗ T_{β}:

{S_{−α}, S_{−β}}φ_{θ}(α, z)φ_{θ}(β, w)
= S_{−α−β}C_{θ}(α, β) (φ_{θ}(−β, z − w)φ_{θ}(α + β, z) − φ_{θ}(α, z − w)φ_{θ}(α + β, w)) .

The same Fay identity (A.21) implies that

φ_{θ}(α, z)φ_{θ}(β, w) = φ_{θ}(−β, z − w)φ_{θ}(α + β, z) − φ_{θ}(α, z − w)φ_{θ}(α + β, w) .

Thus we come to (3.1). □

3. The hierarchy of the Lax equations The Lax operator L^{rot} (3.6) has the following properties:

i) L^{rot} is an sl(N, ℂ)-valued meromorphic function on E_{τ} with a simple pole at the origin satisfying

Res L^{rot}(z)|_{z=0} = ∑_{α} S_{−α}T_{α} .

ii) L^{rot} satisfies the quasi-periodicity conditions

L^{rot}(z + 1) = QL^{rot}(z)Q^{−1} , L^{rot}(z + τ) = ΛL^{rot}(z)Λ^{−1} .

These properties imply that T r(L(z))^{k} are doubly periodic functions with the poles up to the order k. Thereby, they can be expanded in the basis of the Weierstrass function and its derivatives

tr(L^{rot}(z))^{k} = I_{0,k} + I_{2,k}℘(z) + … + I_{k,k}℘^{(k−2)}(z) .  (3.11)

In particular, in this way we obtain the Hamiltonian

tr(L^{rot}(z))^{2} = I_{0,2} + I_{2,2}℘(z) , I_{0,2} = 2 (\frac{i}{2πθ} )^{2} H , I_{2,2} = tr S^{2} .

Proposition 3.3 The coefficients I_{s,k} are in involution with respect to the linear bracket (3.1)

{I_{s,k}, I_{m,j}}_{1} = 0 .  (3.12)

Proof We have

{tr(L_{1}^{rot}(z))^{k}, tr(L_{2}^{rot}(w))^{j}}_{1} = tr{(L_{1}^{rot}(z))^{k}, (L_{2}^{rot}(w))^{j}}_{1} .

Then, it follows from (3.10) that these functionals Poisson commute. Using the expansion (3.11) we arrive at the involutivity of the coefficients (3.12).□

In particular, all functions I_{s,k} Poisson commute with the Hamiltonian H (3.2). Therefore, they play the role of conservation laws of elliptic rotator hierarchy on GL(N, ℂ). We have a tower of \frac{N(N+1)}{2} independent integrals of motion

I_{0,2} I_{2,2}
I_{0,3} I_{2,3} I_{3,3}
. . . . . . . . . . . .
I_{0,n} I_{2,N} . . . . . . I_{N,N}

Note that I_{k,k}, k = 0, 2, 3 …, N are the Casimirs corresponding to the coadjoint orbit

R^{rot} = {S ∈ gl(N, ℂ), S = g^{−1}S^{(0)}g} .

The conservation laws I_{s,k} generate commuting flows on R^{rot}

∂_{s,k}S = {I_{s,k}, S}_{1} , (∂_{s,k} := ∂_{t_{s,k}}) ,

that we call ER_{N} hierarchy.

In what follows we will need another set of the conservation laws coming from the coefficients of the spectral curve

C : F (λ, z) ≡ det(λ · Id + L^{rot}(z)) = 0 .  (3.13)

Here

F (λ, z) = λ^{N} + λ^{N−2}a_{2}(z) + … + a_{N}(z) ,
a_{k}(z) = J_{0k} + J_{2k}℘(z) + … + J_{kk}℘^{(k−2)}(z) .  (3.14)

Since a_{k}(z) and T r(L(z))^{k} are related by the Newton formula, one can recursively represent J_{s,k} as polynomials of I_{m,j}, j ≤ k, m ≤ j

J_{s,k} = ∑ a_{l⃗,m⃗,j⃗}^{s,k} ∏ ; ∑ j_{i}m_{i}=k, l_{i}≤j_{i}(I_{l_{i},j_{i}})^{m_{i}} , (l⃗ = (l_{1}, l_{2}, …)) .  (3.15)

For example, J_{s,2} = −I_{s,2} , (s = 0, 2) ; J_{s,3} = −I_{s,3} , (s = 0, 2, 3) ;

J_{0,4} = \frac{3}{2}I_{0,4} − I_{0,2}^{2} − \frac{g_{2}}{12}I_{2,2}^{2} , (g_{2} = 60 ∑_{m,n}^{′}(m + nτ)^{−4}) ;  (3.16)

J_{2,4} = \frac{3}{2}I_{2,4} − 2I_{0,2}I_{2,2} ; J_{3,4} = \frac{3}{2}I_{3,4} J_{4,4} = \frac{3}{2}I_{4,4} − I_{2,2}^{2} ,

where in the second line we have exploited the relation 12℘^{2} = 2℘′′ + g_{2}.

3.2 The classical GL(N, ℂ) Feigin-Odesski algebras

Here we consider the Feigin-Odesski Poisson brackets related to GL(N, ℂ). In notations of Ref. [5] it is the classical limit of the quadratic associative algebra Q_{N^{2},k}.

It turns out that the same r-matrix (3.8) defines a quadratic Poisson algebra. We start with the Lax operator L^{rot} (3.6). Modify it in the following way

L(z) = −S_{0}Id + L^{rot}(z) ,  (3.17)

where S_{0} commute with S_{α} with respect to the linear brackets (3.1). Define the brackets between the entries of L as follows

{L_{1}(z), L_{2}(w)}_{2} = [r_{N,θ,τ} (z − w), L_{1}(z) ⊗ L_{2}(w)] ,  (3.18)

(see Ref. [13, 14]). These brackets are Poisson, since the Jacobi identity is provided by the classical Yang-Baxter equation (3.9). A finite-dimensional quadratic Poisson algebra is extracted from (3.18) by eliminating the dependence on the spectral parameters z, w.

Proposition 3.4 The quadratic Poisson algebra on gl(N, ℂ)^{∗} has the form

{S_{α}, S_{0}}_{2} = ∑_{γ≠α} S_{α−γ}S_{γ}(℘_{θ}(γ) − ℘_{θ}(α − γ))C_{θ}(α, γ) ,  (3.19)

{S_{α}, S_{β}}_{2} = S_{0}S_{α+β}C_{θ}(α, β) + ∑_{γ≠α,−β} S_{α−γ}S_{β+γ}f_{θ}(α, β, γ)C_{θ}(γ, α − β) ,  (3.20)

where C_{θ}(α, β) are the sl(N, ℂ) structure constants (B.8),

f_{θ}(α, β, γ) = ζ_{θ}(γ) + ζ_{θ}(β − α + γ) − ζ_{θ}(β + γ) + ζ_{θ}(α − γ)
= −\frac{ϑ′(0)ϑ_{θ}(α)ϑ_{θ}(β)ϑ_{θ}(β − α + 2γ)}{ϑ_{θ}(α − γ)ϑ_{θ}(β + γ)ϑ_{θ}(β − α + γ)ϑ_{θ}(γ)} ,  (3.21)

and ϑ_{θ}, ζ_{θ} are the zeta constants (B.10), (B.11).

Proof. The equation (3.18) yields the following form of the brackets

{S_{a}, S_{b}}φ_{θ}(a, z)φ_{θ}(b, w) = ∑_{γ} S_{a−γ}S_{b+γ}φ_{θ}(γ, z −w)φ_{θ}(a−γ, z)φ_{θ}(b+γ, w)C_{θ}(γ, a−b) . (3.22)

Note first, that the common exponents in the expressions of the functions φ_{θ} (B.13) coincide in the left and the right sides. This allows one to pass to the functions ϕ_{θ} and in this way to use the Fay-type identities. We rewrite the last expression in the form

{S_{a}, S_{b}}ϕ_{θ}(z)ϕ_{θ}(w) = ∑_{γ} S_{a−γ}S_{b+γ}C_{θ}(γ, a − b)

× (ϕ_{θ}(γ, z − w)ϕ_{θ}(a − γ, z)ϕ_{θ}(b + γ, w) − ϕ_{θ}(a − b − γ, z − w)ϕ_{θ}(b + γ, z)ϕ_{θ}(a − γ, w)) .

Consider first the case b = 0. Then we can apply (A.30) for u = (a_{1} + a_{2}τ)θ, v = (γ_{1} + γ_{2})θ. It immediately leads us to (3.19). If a, b ≠ 0 we have the terms of two types. For γ ≠ a, −b one uses (A.27), (A.28) and finds the second term in the right side of (3.20). If γ = a or γ− = b one should take into account that ϕ_{θ}(0, z) = 1 and then use the Fay identity (A.21). This gives us the first term. The identity (3.21) follows from (A.29). □

We denote by P_{N,θ,τ} the quadratic Poisson algebra (3.19), (3.20). Recall that τ is the modular parameter of the auxiliary curve E_{τ} and θ = k/N. The algebra P_{2, \frac{1}{2} ,τ} is the classical Sklyanin algebra [14]. In this case the r.h.s. of (3.20) contains only the first term.

As for the linear brackets, there exist N Casimirs C_{0}^{(2)}, C_{2}^{(2)}, …, C_{N}^{(2)} of the algebra P_{N,θ,τ} [18]. They can be read off from the expansion of the spectral curve (3.13) in the basis of elliptic functions

det(S_{0} · Id + L^{rot}(z)) = C_{0}^{(2)} + ∑_{k=2}^{N} C_{k}^{(2)}℘^{(k−2)}(z) .  (3.23)

It follows from (3.13), (3.14) and (3.17) that they are linear combinations of the integrals J_{s,k}

C_{0}^{(2)} = S_{0}^{N} + ∑_{m=2}^{N} S_{0}^{N−m}J_{0,m} ,  (3.24)

C_{k}^{(2)} = ∑_{m=0}^{N−k} S_{0}^{N−k−m}J_{k,k+m} , (k = 2, …, N) .  (3.25)

3.3 The bihamiltonian structure

Two Poisson structures are called compatible (or, form a Poisson pair) if their linear combinations are Poisson structures as well.

Proposition 3.5 The linear and quadratic Poisson brackets on gl(N, ℂ) are compatible.

Proof Let us replace S_{0} → S_{0}+λ in the quadratic brackets (3.19), (3.20). Then (3.19) does not change, while (3.20) acquires an additional linear term. In this way we define a one-parameter family of the brackets

{S, S}_{λ} := {S, S}_{2} + λ{S, S}_{1} ,  (3.26)

where {S, S}_{1} denote the linear brackets (3.1). Therefore, the linear combination {S, S}_{λ} of two brackets can be obtained from the quadratic bracket by a simple shift of S_{0}. The result of the shift, of course, still satisfies the Jacobi identity, and hence it is a Poisson bracket for any λ. □

We denote this family of the quadratic Poisson algebras by P_{N,θ,τ,λ}. The algebras are isomorphic for different λ and degenerate to the linear Poisson algebra gl(N, ℂ)^{∗} as λ → ∞.

Consider the Casimir functions of the Poisson algebra P_{N,θ,τ,λ}

h_{k}(λ) = C_{k}^{(2)}(S_{0} + λ) (k = 0, 2, …, N) .

It follows from (3.24) and (3.25) that h_{k}(λ) are polynomials in λ

h_{k}(λ) = ∑_{m=0}^{N−k} h_{m,k}λ^{m} .  (3.27)

The coefficients are defined as

h_{a,k} = \frac{1}{a!}∂_{λ}^{(a)}C_{k}^{(2)}(S_{0} + λ)_{λ=0} .

It implies that

h_{a,0} = \frac{N!}{(N − a)!a!}S_{0}^{N−a} + ∑_{m=2}^{N−a} \frac{(N − m)!}{(N − m − a)!a!}S_{0}^{N−m−a}J_{0,m} ,  (3.28)

h_{a,k} = ∑_{m=0}^{N−k−a} \frac{(N − k − m)!}{(N − k − m − a)!a!}S_{0}^{N−k−m−a}J_{k,k+m} , (k = 2, …, N) .  (3.29)

In particular,

h_{N−1,0} = NS_{0} , h_{N−2,0} = \frac{N(N − 1)}{2}S_{0}^{2} + J_{0,2} .  (3.30)

Conversely, one can express J_{m,k+m} as a linear combination of h_{N−m−s,m}, s = 0, …, k from the relations (3.28) and (3.29). For example,

J_{j,j} = h_{N−j,j} , J_{j,j+1} = h_{N−j−1,j} − (N − j)S_{0}h_{N−j,j} ,

J_{j,j+2} = h_{N−j−2,j} − (N − j)S_{0}h_{n−j−1,j} + \frac{(N − j)(N − j − 1)}{2}S_{0}^{2}h_{N−j,j} .

We arrange the quantities h_{m,k} in the triangular tableau

h_{0,0} h_{1,0}  …  . . . . . . . . . . . . . . .  …  h_{N−1,0} h_{N,0}
h_{0,2}  h_{1,2}  . . . . . . . . .  h_{N−3,2}  h_{N−2,2}
h_{0,3}  h_{1,3} . . . . . . . . . h_{N−4,4}  h_{N−3,3}
…  . . . . . . . . .  …
h_{0,N−1}  h_{1,N−1}
h_{0,N}

(3.31)

Note, that the second line corresponding to h_{s,1} is absent since h_{1}(λ) = 0. The left side of the triangle contains the Casimirs h_{0,j} = C_{j}^{(2)} (3.24), (3.25) of P_{N,θ,τ}, while the right side represents the Casimirs h_{N−j,j} = J_{j,j} of the linear brackets on gl(N, ℂ)^{∗}.

The remarkable property of the quantities h_{a,k} is that they are in involution with respect to both the linear and quadratic brackets:

{h_{a,k}, h_{b,j}}_{1,2} = 0 ,

see, e.g., [19]. This fact follows from the identities

{h_{k}(λ), h_{j}(λ)}_{λ} = 0 ,  (3.32)

held for each λ.

Proposition 3.6 The integrals h_{a,k} provide the recurrence relation for the elliptic rotator hierarchy ER_{N}

{h_{a,k}, S}_{1} = −{h_{a+1,k}, S}_{2} .  (3.33)

Proof. Since h_{k}(λ) are the Casimirs, we have {h_{k}(λ), S}_{λ} = 0. Substitute the representation (3.27) into this equation. Then the recurrence relation (3.33) comes from the coefficient in front of λ^{a}. □

In this way one can start with the Casimirs of the linear brackets and produce a non-trivial dynamical system using the quadratic brackets

{J_{k,k}, S}_{2} = −{J_{k,k+1}, S}_{1} ,  (3.34)

(see (3.30)). In particular, the flow (3.4) corresponding to H = I_{0,2} can be represented by the quadratic brackets with S_{0}:

∂_{t}S = {S_{0}, S}_{2} .

Alternatively, one can start with the Hamiltonians in the left side of the table (3.31) and the linear brackets.

This allows us to conclude the proof of Theorem 2.1 for the hierarchy ER_{N}. Namely, Propositions 3.2, 3.4, 3.5, and 3.6 are equivalent, respectively, to i), ii), iii), and iv).

4 Elliptic Rotators on non-commutative torus

4.1 The Lax equation and integrals of motion

1. The system description The elliptic rotator on the non-commutative torus A_{θ} is a generalization of the elliptic rotator on GL(N, ℂ). We consider the Euler-Arnold top on the group SIN_{θ} of the NCT algebra A_{θ} (Appendix C). The Lie coalgebra sin_{θ}^{∗} is equipped with the linear Poisson brackets

{S_{α}, S_{β}}_{1} = C_{θ}(α, β)S_{α+β} , (α ∈ ℤ̃^{(2)}) ,  (4.1)

where S = ∑ S_{−α}T_{α} and

ℤ̃^{(2)} = {α = (α_{1}, α_{2}) , α_{j} ∈ ℤ , α ≠ (0, 0)} .

The inverse inertia tensor J maps sin_{θ}^{∗} → sin_{θ}, and depends on four parameters: θ, τ ∈ ℂ, (ℑmτ > 0) and ϵ = (ϵ_{1}, ϵ_{2}), where 0 < θϵ_{a} ≤ 1 and ϵ_{a}θ are irrational numbers. The components of J are given by the elliptic constants J_{α} = ℘_{θ,ϵ}(α) (C.18)

J(S) = ∑_{α∈ℤ̃^{(2)}} ℘_{θ,ϵ}(α)S_{−α}T_{α} .

The Hamiltonian has the form

H = 2π^{2}θ^{2} ∫_{A_{θ}} S · J(S) = −\frac{1}{2} ∑_{α∈ℤ̃^{(2)}} ℘_{θ,ϵ}(α)S_{α}S_{−α} .  (4.2)

Introduce a complex structure on the NCT A_{θ} depending on ϵ = (ϵ_{1}, ϵ_{2}) and τ such that for X = ∑_{a} c_{a}T_{a} one has:

¯∂_{ϵ,τ} X = ∑_{a}(ϵ_{1}a_{1} + ϵ_{2}a_{2}τ)c_{a}T_{a} .  (4.3)

Then, in the Moyal representation, the operator J can be identified with the pseudo-differential operator

J(S)(x) = ℘(θ ¯∂_{ϵ,τ})S(x) ,  (4.4)

and

H = 2π^{2}θ^{2} ∫_{A_{θ}} S · ℘(θ ¯∂_{ϵ,τ})S .  (4.5)

The equation of motion in the form of the Moyal brackets has the standard form

∂_{t}S = {S, (℘(θ ¯∂_{ϵ,τ})S)}^{θ} := ad_{J(S)}^{∗}S ,  (4.6)

or, in the components,

∂_{t}S_{α} = ∑_{γ∈ℤ̃^{(2)}} S_{γ}℘_{θ,ϵ}(γ)S_{α−γ} .  (4.7)

2. The r-matrix and the Lax equation Define formally the Lax operator

L^{rot}(z) = − ∑_{α∈ℤ̃^{(2)}} S_{α}φ_{θ,ϵ}(α, z)T_{α} ,  (4.8)

where φ_{θ,ϵ} is given by (C.19). Below we formulate the conditions on the phase space, which show that L^{rot}(z) is well defined for z ∈ E_{τ} , z ≠ 0. Note that

∫_{A_{θ}} L^{rot}(z) = 0 .

The following Proposition is an infinite-dimensional analog of Proposition 3.1.

Proposition 4.1 The equations of motion (4.7) have the Lax form

∂_{t}L^{rot} = [L^{rot}, M^{rot}] := {L^{rot}, M^{rot}}^{θ}  (4.9)

with

M^{rot}(z) = − ∑_{α∈ℤ̃^{(2)}} S_{α}f_{θ,ϵ}(α, z)T_{α} ,  (4.10)

and f_{θ,ϵ} (C.21).

The proof is analogous to the finite-dimensional case. It is based on the Calogero functional equation (A.22).

Introduce the classical r-matrix on sin_{θ} ⊗ sin_{θ} as the following sum:

r_{θ,ϵ,τ} (z − w) = ∑_{γ∈ℤ̃^{(2)}} φ_{θ,ϵ}(γ, z − w)T_{γ} ⊗ T_{−γ} ,  (4.11)

Lemma 3.1 and Proposition 3.2 have the following analogs for the NCT:

Proposition 4.2 a) The r-matrix (4.11) satisfies the classical Yang-Baxter equation (3.9).

b) The r-matrix defines the Poisson brackets for the entries of the Lax operator (4.8)

{L_{1}^{rot}(z), L_{2}^{rot}(w)}_{1} = [r_{θ,ϵ,τ} (z − w), L^{rot}(z) ⊗ Id + Id ⊗ L^{rot}(w)] .  (4.12)

They are equivalent to the linear brackets (4.1) on sin_{θ}^{∗}.

The proof of Proposition 4.2(a, b) is similar to those of Lemma 3.1 and Proposition 3.2 and is based on the Fay identity (A.21).

3. The hierarchy of the Lax equations In order to construct integrals of motion we first discuss the properties of the Lax operator. This operator is the meromorphic quasi-periodic function on E_{τ} with a simple pole

Res L^{rot}(z)|_{z=0} = S ,
L^{rot}(z + 1) = U_{1}^{ϵ_{1}}L^{rot}(z)U_{1}^{−ϵ_{1}} , L^{rot}(z + τ) = U_{2}^{ϵ_{2}}L^{rot}(z)U_{2}^{−ϵ_{2}} .

where U_{1}, U_{2} are the generators of NCT A_{θ} (C.1). (This follows from (4.8), (A.9) and (B.15).) It means that the integrals

∫_{A_{θ}}(L^{rot}(z))^{j}

are doubly periodic functions on E_{τ} with poles up to order j. Thereby, we have the expansion

∫_{A_{θ}}(−L^{rot}(z))^{j} = I_{0,j} + ∑_{r=2}^{j} I_{r,j}℘^{(r−2)}(z) .  (4.13)

In particular,

∫_{A_{θ}}(L_{θ})^{2}(z) = I_{0,2} + ℘(z) ∫_{A_{θ}} S^{2} , where I_{0,2} = 2 (\frac{i}{2πθ} )^{2} H .

Note that

I_{j,j} = ∫_{A_{θ}} S^{j}

are the Casimirs of the linear brackets.

The phase space R of the elliptic rotator is determined by the following properties of the integrals I_{s,j}

R = \{S ∈ A_{θ}^{∗} | i) I_{s,j} < ∞ , ii) lim_{j→∞} \frac{I_{s,j+1}(S)}{I_{s,j}(S)} < 1 for all s ≤ j\} .  (4.14)

The first condition means that the traces (4.13) of the Lax operator are well defined on E_{τ} for z ≠ 0. In particular, we have

I_{0,2} < ∞ , i.e. ∑ ℘_{θ}(α)S_{α}S_{−α} < ∞ .  (4.15)

We will use the second condition below to define the Casimir functions for the family of quadratic Poisson brackets.

Further, due to Lemma 3.1, an analog of Proposition 3.2 is the following.

Proposition 4.3 The quantities I_{s,k} are pairwise Poisson commute.

The conservation laws I_{s,k} generate commuting flows on the phase space R with respect to the linear brackets

∂_{s,k}S = {I_{s,k}, S}_{1}^{θ} ,  (4.16)

where ∂_{s,k} stands for the corresponding time derivative ∂_{t_{s,k}}. We call these equations ER_{θ} hierarchy on the NCT.

One can show that all flows can be represented in the Lax form with L^{rot} (4.8) and the corresponding M_{s,k}. Furthermore, there exists a set of the integrals of J_{s,k}-type. They are related to the integrals I_{m,j} by the same formulae (3.15), (3.16).

4.2 Quadratic Poisson algebras and the bihamiltonian structure on NCT

To define the quadratic Poisson algebra on the phase space R (4.14) we modify the Lax operator (4.8):

L = −S_{0} · T_{0} + L^{rot} = S_{0} · T_{0} + ∑_{α} S_{−α}φ_{θ,ϵ}(α, z)T_{α} ,  (4.17)

where S_{0} commutes with S_{α} with respect to the linear brackets. Due to the YB equation, the r-matrix (4.11) defines the quadratic Poisson brackets

{L_{1}(z), L_{2}(w)}_{2}^{θ} = [r_{θ,ϵ,τ} (z − w), L_{1}(z) ⊗ L_{2}(w)] .  (4.18)

Again, we can get rid of the dependence on the spectral parameter.

Proposition 4.4 The quadratic Poisson brackets on the phase space R are defined as follows

{S_{α}, S_{0}}_{2}^{θ} = ∑_{γ≠α} S_{α−γ}S_{γ}(℘_{θ,ϵ}(γ) − ℘_{θ,ϵ}(α − γ))C_{θ}(α, γ) ,  (4.19)

{S_{α}, S_{β}}_{2}^{θ} = S_{0}S_{α+β}C_{θ}(α, β) + ∑_{γ≠α,−β} S_{α−γ}S_{β+γ}f_{θ,ϵ}(α, β, γ)C_{θ}(γ, α − β) ,  (4.20)

where C_{θ}(α, γ) is a structure constant of the sin_{θ}-algebra,

f_{θ,ϵ}(α, β, γ) = ζ_{θ,ϵ}(γ) + ζ_{θ,ϵ}(β − α + γ) − ζ_{θ,ϵ}(β + γ) + ζ_{θ,ϵ}(α − γ)  (4.21)

= −\frac{ϑ′(0)ϑ_{θ,ϵ}(α)ϑ_{θ,ϵ}(β)ϑ_{θ,ϵ}(β − α + 2γ)}{ϑ_{θ,ϵ}(α − γ)ϑ_{θ,ϵ}(β + γ)ϑ_{θ,ϵ}(β − α + γ)ϑ_{θ,ϵ}(γ)} ,

and ϑ_{θ,ϵ}, ζ_{θ,ϵ} are the constants (C.17), (C.18).

Proof The equality (4.18), being reduced to the coefficients in the front of T_{a} ⊗ T_{b}, can be rewritten in the form (3.22) where the functions φ_{θ}(a, z) are replaced on φ_{θ,ϵ}(a, z). Then using the cubic functional equations (A.27) and (A.28) we obtain to the algebra (4.19), (4.20). Due the definition of the phase space (4.15) the series in the right hand sides of (4.19), (4.20) converge. □

We denote this Poisson algebra by P_{θ,ϵ,τ}. Set now ϵ_{1} = ϵ_{2} = 1. The corresponding algebra P_{θ,1,τ} can be considered as a special large N limit of the finite-dimensional algebras P_{N,θ,τ} (3.19), (3.20). For this we just replace the rational number θ = k/N by an arbitrary irrational number 0 ≤ θ < 1 in the algebra P_{N,θ,τ}.

Proposition 4.5 a) The linear (4.1) and quadratic (4.19), (4.20) Poisson brackets defined on R are compatible.

b) The ER_{θ} hierarchy admits the bihamiltonian structure.

Proof. One can shift S_{0} + λ and define the family Poisson algebras P_{θ,ϵ,τ,λ}. To define the Casimir elements on P_{N,θ,τ,λ} we consider

log det(S_{0} + λ + L^{rot}(z)) = log(S_{0} + λ) + ∑_{k=2}^{∞} \frac{(−1)^{k+1}}{k(S_{0} + λ)^{k}} trL(z)^{k} .

Then we come to the infinite set of Casimirs ^{1}

C̃_{0}^{(2)}(S_{0} + λ) = log(S_{0} + λ) + ∑_{k=2}^{∞} \frac{(−1)^{k+1}}{k(S_{0} + λ)^{k}} I_{0,k} ,

C̃_{j}^{(2)}(S_{0} + λ) = ∑_{k=2}^{∞} \frac{(−1)^{k+1}}{k(S_{0} + λ)^{k}} I_{j,k} , (j = 2, 3, …) .

These functionals are well-defined on the phase space (4.14). This allows us to introduce the new set of conserved quantities

C̃_{0}^{(2)}(S_{0} + λ) = ∑_{s} h̃_{s,0}λ^{s} ,

C̃_{j}^{(2)}(S_{0} + λ) = ∑_{s} h̃_{s,j}λ^{s} .

The latter Poisson commute with respect to the both types of brackets and give rise to the bihamiltonian structure of the hierarchy

{h̃_{s,j}, S}_{1}^{θ} = −{h̃_{s+1,j}, S}_{2}^{θ} .  (4.22)

In particular, we can represent the flow (4.6) in the form

∂_{t}S = {S_{0}, S}_{2}^{θ} .

(see (3.30)). □

This proposition concludes the proof of Theorem 2.1 for the hierarchy ER_{θ}.

5 Elliptic Rotators on SDiff(T^{2})

5.1 Description of the hierarchy

In the ”dispersionless” limit θ → 0 the Lie algebra sin_{θ} turns to the Lie algebra Ham(T^{2}) of Hamiltonians on a two-dimensional torus, see (C.22). This algebra of Hamiltonian functions can be represented by the Lie algebra of the corresponding divergence-free vector fields SV ect(T^{2}). More precisely, to pass from Ham(T^{2}) to SV ect(T^{2}) one has to discard the constant Hamiltonians, but add the “flux” vector fields ∂/∂x_{1} and ∂/∂x_{2} corresponding to multivalued Hamiltonian functions x_{1} and x_{2} on the torus.

We define the elliptic rotator system ER on the Lie group SDiff(T^{2}) by considering the limit θ → 0 of the ER_{θ}-system described above. Let θ → 0, ϵ_{1,2} → ∞, such that

lim_{θ→0}(θϵ_{1,2}) = ϵ′_{1,2} < 1 , ϵ′_{1,2} are irrational .  (5.1)

In what follows we drop the superscript ′.

Let S = ∑_{α} S_{−α}e(α·x) ∈ Ham^{∗}(T^{2}), where e(α·x) is the Fourier basis (C.23) of Ham^{∗}(T^{2}). In the Fourier basis, the linear Poisson bracket assumes the form

{S_{α}, S_{β}}_{1} = (α × β)S_{α+β} , α × β = α_{1}β_{2} − α_{2}β_{1} .  (5.2)

1Since we pass from det to log det the basis of the Casimir functions differs from (3.24)

The inverse inertia operator J : Ham^{∗}(T^{2}) → Ham(T^{2}) becomes

J : S_{α} → ℘_{ϵ}(α)S_{α} , ℘_{ϵ}(α) = ℘(ϵ_{1}α_{1} + ϵ_{2}α_{2}τ; τ) ,

where α ∈ ℤ̃^{(2)} (B.5). The operator is well defined since ϵ_{j} are irrational. In other words, the operator J is the pseudo-differential operator

J : S(x) → ℘(¯∂_{ϵ,τ})S(x) ,

where ¯∂_{ϵ,τ} is the operator of the complex structure on the elliptic curve, commutative torus T^{2}. In fact, the complex structure depends on the ratio τϵ_{2}/ϵ_{1}.

The quadratic Hamiltonian of the system is

H = −\frac{1}{2} ∑_{γ} S_{γ}℘_{ϵ}(γ)S_{−γ} = −\frac{1}{2} ∫_{T^{2}} S(℘(¯∂_{ϵ,τ})S) .  (5.3)

and the corresponding equations of motion are

∂_{t}S = {S, ℘(¯∂_{ϵ,τ})S}_{1} .  (5.4)

This equation is an elliptic analog of the hydrodynamics equation on the torus T^{2}, regarded as an elliptic curve, see Remark below. We call it the elliptic hydrodynamics.

Define the Lax operator

L^{rot}(x; z) = − ∑_{α∈ℤ̃^{(2)}} S_{α}φ_{ϵ}(α, z)e(α · x) , φ_{ϵ}(α, z) = φ(ϵ_{1}α_{1} + τϵ_{2}α_{2}, z) .  (5.5)

The conditions on the phase space formulated below, see (5.12), ensure that the operator L^{rot}(x, z) is well defined for z ∈ E_{τ} , z ≠ 0. Note that

∫_{T^{2}} L_{ϵ}^{rot}(z) = 0 .

Proposition 5.1 The equations of motion (5.4) have the dispersionless Lax representation

L^{rot} = {L^{rot}, M^{rot}}_{1}

with L^{rot} given by (5.5) and

M^{rot}(x; z) = − ∑_{α∈ℤ̃^{(2)}} S_{α}f_{ϵ}(α, z)e(α · x) ,  (5.6)

where

f_{ϵ}(α, z) = f(ϵ_{1}α_{1} + τϵ_{2}α_{2}, z) = e(ϵ_{2}α_{2})∂_{u}ϕ(u, z)|_{u=ϵ_{1}α_{1}+τϵ_{2}α_{2}} .

The proof of Proposition 5.1 is the same as Proposition 3.1.

Note that operators L^{rot} and M^{rot} satisfy the quasi-periodicity properties

L^{rot}(x_{1}, x_{2}; z) = L^{rot}(x_{1} + ϵ_{2}, x_{2}; z + 1) ,  (5.7)
L^{rot}(x_{1}, x_{2}; z) = L^{rot}(x_{1}, x_{2} − ϵ_{1}; z + τ) ,  (5.8)
M^{rot}(x_{1}, x_{2}; z) = M^{rot}(x_{1} + ϵ_{2}, x_{2}; z + 1) ,  (5.9)

and

M^{rot}(x_{1}, x_{2}; z) − M^{rot}(x_{1}, x_{2} − ϵ_{1}; z + τ) = 2πiL^{rot}(x_{1}, x_{2}, z) .  (5.10)

Furthermore, it follows from (5.7) and (5.8) that there exists the expansion

∫_{T^{2}}(L^{rot}(x, z))^{k} = I_{0,k} + ∑_{s=2}^{k} I_{s,k}℘^{(s−2)}(z) .  (5.11)

The phase space R^{0} of the elliptic rotator on SDiff(T^{2}) can be described similarly to the phase space R for SIN_{θ}-group, see (4.14):

R^{0} = \{S ∈ Ham^{∗}(T^{2}) | i) I_{s,j} < ∞ , ii) lim_{j→∞} \frac{I_{s,j+1}(S)}{I_{s,j}(S)} < 1 for all s ≤ j\} .  (5.12)

Define the r-matrix on T^{2} ⊗ T^{2} by

r_{ϵ,τ} (x, y; z) = ∑_{α∈ℤ̃^{(2)}} φ_{ϵ}(α, z)e(α · x)e(−α · y) , z ∈ E_{τ} .  (5.13)

Proposition 5.2 a) The r-matrix (5.13) satisfies the ”dispersionless” YB equation

{r_{ϵ,τ} (x, y; z − w), r_{ϵ,τ} (x, v; z)} + {r_{ϵ,τ} (x, y; z − w), r_{ϵ,τ} (y, v; w)}
+{r_{ϵ,τ} (x, v; z), r_{ϵ,τ} (y, v; w)} = 0 .  (5.14)

b) In terms of the Lax operator (5.5) the canonical brackets on T^{2} take the form

{L^{rot}(x; z), L^{rot}(y; w)} = {r_{ϵ,τ} (x, y, z − w), L^{rot}(x; z)} + {r(x, y, z − w)L^{rot}(y; w)} . (5.15)

Proof These statements are consequences of the Fay identity (A.21), cf. Proposition 4.2.□

The form of the brackets (5.15) implies that

{∫_{T^{2}}(L^{rot}(x, z))^{k}, ∫_{T^{2}}(L^{rot}(x, z))^{j}} = 0 .

In turn, then (5.11) produces the infinite sequence of conservation laws I_{s,k} in involution. They define the hierarchy of commuting flows

∂_{s,k}S = {I_{s,k}, S}_{1} ,  (5.16)

which is the dispersionless limit of the hierarchy (4.16). The equations can be represented in the form of the dispersionless Lax equations.

5.2 Quadratic Poisson algebras, and the bihamiltonian structure on SDiff(T^{2})

To describe the quadratic Poisson brackets, we first pass from the Lie algebraic Lax operator L^{rot} to the operator

L_{ϵ} = −S_{0} + L^{rot}(x; z) .  (5.17)

Define the quadratic Poisson algebra by the formula

{L_{ϵ}(x; z), L_{ϵ}(y; w)} = {r_{ϵ,τ} (x, y; z − w), L_{ϵ}(x; z)L_{ϵ}(y; w)} .  (5.18)

Proposition 5.3 In terms of the Fourier modes, the Poisson algebra (5.18) has the form

{S_{α}, S_{0}}_{2} = ∑_{γ≠α} S_{α−γ}S_{γ}(℘_{ϵ}(γ) − ℘_{ϵ}(α − γ))(α × γ) ,  (5.19)

{S_{α}, S_{β}}_{2} = S_{0}S_{α+β}(α × β) + ∑_{γ≠α,−β} S_{α−γ}S_{β+γ}f_{ϵ}(α, β, γ)(γ × (α − β)) ,  (5.20)

where

f_{ϵ}(α, β, γ) = ζ_{ϵ}(γ) + ζ_{ϵ}(β − α + γ) − ζ_{ϵ}(β + γ) + ζ_{ϵ}(α − γ)

= −\frac{ϑ′(0)ϑ_{ϵ}(α)ϑ_{ϵ}(β)ϑ_{ϵ}(β − α + 2γ)}{ϑ_{ϵ}(α − γ)ϑ_{ϵ}(β + γ)ϑ_{ϵ}(β − α + γ)ϑ_{ϵ}(γ)} ,

ζ_{ϵ}(γ) = ζ(ϵ_{1}γ_{1} + τϵ_{2}γ_{2}), ϑ_{ϵ}(γ) = ζ(ϵ_{1}γ_{1} + τϵ_{2}γ_{2}).

The proof of this Proposition is similar to that of Proposition 3.2.

By the same trick as before we can show that the linear and the quadratic brackets are compatible. It allows us to construct the integral of the h̃_{s,k} type and define the recurrence representation (4.22) for the bihamiltonian structure on the hierarchy. In particular, the equation (5.4) can be represented as

∂_{t}S = {S_{0}, S}_{2} .

5.3 Rational limit of the elliptic hydrodynamics

1. Description of the limit So far we have been dealing with the elliptic curve E_{τ} parameterized by the two half-periods ω_{1}, ω_{2}, τ = ω_{2}/ω_{1}. WE replace our main functions in the following way:

ζ_{ϵ}(γ) = ζ(ϵ_{1}γ_{1}ω_{1} + ϵ_{2}γ_{2}ω_{2}; ω_{1}, ω_{2},) , ℘_{ϵ}(γ) = ℘(ϵ_{1}γ_{1}ω_{1} + ϵ_{2}γ_{2}ω_{2}; ω_{1}, ω_{2},) .  (5.21)

Now we are going to consider the rational limit of the elliptic curves, i.e., degeneration lim ω_{1,2} → ∞. We look at the double scaling limit

lim ϵ_{1,2} → 0 , lim ω_{1,2} → ∞ , lim ϵ_{1}ω_{1} = 1 , lim ϵ_{2}ω_{2} = τ .

Then lim ω_{1,2} → ∞ leads to the rational degeneration of the elliptic functions (5.21)

lim_{ω_{1,2}→∞} ζ_{ϵ}(γ) = \frac{1}{γ_{τ}} , lim_{ω_{1,2}→∞} ℘_{ϵ}(γ) = \frac{1}{γ_{τ}^{2}} , γ_{τ} = γ_{1} + γ_{2}τ

This implies, in particular, that the inverse inertia tensor assumes the form

J(S) = ¯∂^{−2}S(x_{1}, x_{2}) = ∑_{α∈ℤ̃^{(2)}} \frac{1}{α_{τ}^{2}} S_{−α}T_{α} ,  (5.22)

where α_{τ} = α_{1} + α_{2}τ. The equations of motion are defined by the corresponding quadratic Hamiltonian

H = −\frac{1}{2} ∫_{T^{2}} S¯∂^{−2}S = −\frac{1}{2} ∑_{α∈ℤ̃^{(2)}} \frac{1}{α_{τ}^{2}} S_{α}S_{−α} .  (5.23)

with respect to the linear Poisson bracket on Ham(T^{2}):

∂_{t}S = {S, ¯∂^{−2}S)}_{1} .  (5.24)

In the Fourier components the latter becomes

∂_{t}S_{α} = ∑_{γ∈ℤ̃^{(2)}} S_{γ} \frac{1}{γ_{τ}^{2}} S_{α−γ} .

We call these equations the modified hydrodynamics on the torus T^{2}.

Remark 5.1 Recall that the standard hydrodynamics of an ideal fluid on the torus T^{2} is given by the Euler equation

∂_{t}S = {S, Δ^{−1}S}_{1},

on the vorticity function S. The definition of the Laplace operator uses metric, while to define the modified hydrodynamics (5.24) we need to fix a complex structure on T^{2}.

Now consider the Lax representation for the modified hydrodynamics. In the double scaling limit we have

lim φ_{ϵ}(α, z) = exp(2πiα_{1}z) (\frac{1}{α_{τ}} + \frac{1}{z}) ,

lim f_{ϵ}(α, z) = − exp(2πiα_{1}z)\frac{1}{α_{τ}^{2}}

Therefore, the Lax pair (5.6) in the rational limit is

L(x_{1}, x_{2}; z) = − ∑_{α} S_{α} exp(2πiα_{1}z) (\frac{1}{α_{τ}} + \frac{1}{z}) T_{α} .

M(x_{1}, x_{2}) = − ∑_{α} S_{α} exp(2πiα_{1}z)\frac{1}{α_{τ}^{2}} T_{α} .

We can drop the exponents in the both operators and come to the following expressions

L(x_{1}, x_{2}; z) = −¯∂^{−1}S(x_{1}, x_{2}) − \frac{1}{z}S(x_{1}, x_{2}) ,  (5.25)

and

M(x_{1}, x_{2}) = −¯∂^{−2}S(x_{1}, x_{2}) .  (5.26)

It is easy to see that the dispersionless Lax equation

∂_{t}L(x_{1}, x_{2}; z) = {L(x_{1}, x_{2}; z), M(x_{1}, x_{2})}

is equivalent to the equations (5.24).

The infinite set of the integrals of motion comes from the decomposition

(−1)^{k} ∫_{T^{2}} L^{k}(x_{1}, x_{2}; z)dx_{1}dx_{2} = I_{0,k} + ∑_{s=2}^{k} I_{m,k}z^{−m} .

The corresponding infinite hierarchy of the modified hydrodynamics

∂_{s,t}S = {I_{s,k}, S}_{1}

has the dispersionless Lax form representation.

2. Bihamiltonian structure It turns out that the bihamiltonian structure survives in this limit. In fact, we have the following quadratic Poisson algebra P_{0,τ} on the commutative torus T^{2}

\{S_{α}, S_{0}\}_{2} = ∑_{γ≠α} S_{α−γ}S_{γ}(\frac{1}{γ_{τ}^{2}} − \frac{1}{(α_{τ} − γ_{τ})^{2}} )(α × γ) ,  (5.27)

{S_{α}, S_{β}}_{2} = S_{0}S_{α+β}(α × β) + ∑_{γ≠α,−β} S_{α−γ}S_{β+γ}f(α, β, γ)(γ × (α − β)) ,  (5.28)

where

f(α, β, γ) = (\frac{1}{γ_{τ}} + \frac{1}{β_{τ} − α_{τ} + γ_{τ}} − \frac{1}{β_{τ} + γ_{τ}} + \frac{1}{α_{τ} − γ_{τ}} ) , γ_{τ} = γ_{1} + τγ_{2} .

Similarly to the above, we can consider a one-parametric family of the quadratic algebras by replacing S_{0} → S_{0} + λ. Note that it degenerates to the standard Poisson brackets on T^{2} in the limit λ → ∞. This allows us to define the bihamiltonian structure for the hierarchy of the modified hydrodynamics. In particular, the equation (5.24) can be rewritten in the form as

∂_{t}S = {S_{0}, S}_{2} .

Such a bihamiltonian structure is a curious feature in the modified hydrodynamics, emphasizing its drastic difference from the classical hydrodynamics.

6 Quantum counterparts

In this section we present two associative algebras “quantizing” the discussed above two Poisson algebras in the case of the NCT. It is easy with the linear bracket. Indeed, the quantization of the Lie-Poisson algebra on sin_{θ}^{∗} leads to the universal enveloping algebra sin_{θ}.

Replace the classical variables S on the NCT A_{θ} by non-commuting variables Ŝ. Now we consider the quantization of the quadratic Poisson algebra P_{θ,ϵ,τ} and construct the associative algebra P̂_{θ,ϵ,τ,ℏ}, where ℏ ∈ E_{τ} is the deformation parameter. For this we introduce the quantum R-matrix related to the group SIN_{θ}. The quantum Yang-Baxter equation defined on GL(N, ℂ) [13, 14] is generalized to SIN_{θ} in the following way. The quantum R-matrix on SIN_{θ} ⊗ SIN_{θ} assumes the form

R(z, w) = ∑_{c} φ_{θ,ϵ}(c, z − w|ℏ)T_{c} ⊗ T_{−c} ,

where T_{c} is the basis of SIN_{θ} and the main ingredient in our construction is the function

φ_{θ,ϵ}(c, z|ℏ) = e_{θ}(ϵ_{2}c_{2}θ)ϕ((ϵ_{1}c_{1} + τϵ_{2}c_{2})θ + ℏ, z) , ℏ ∈ E_{τ} .

Note that, in contrast with the classical r-matrix (4.11), there is an additional term φ_{θ,ϵ}(0, z − w|ℏ)T_{0} ⊗ T_{0}.

Proposition 6.1 The matrix R satisfies the quantum YB equation

R_{12}(z − w)R_{13}(z)R_{23}(w) = R_{23}(w)R_{13}(z)R_{12}(z − w) .  (6.1)

Proof. Consider the coefficients in front of T_{a} ⊗ T_{b} ⊗ T_{c} in the l.h.s of (6.1). It vanishes if a + b + c ≠ 0. Therefore (6.1) implies

e_{θ}(a × b + 2b × c)φ_{θ,ϵ}(c, z − w|ℏ)φ_{θ,ϵ}(a − c, z|ℏ)φ_{θ,ϵ}(b + c, w|ℏ)
−e_{θ}(a × b + 2b × f)φ_{θ,ϵ}(f, z − w|ℏ)φ_{θ,ϵ}(a − f, z|ℏ)φ_{θ,ϵ}(b + f, w|ℏ) = 0

Here a, b are fixed and c, f are arbitrary elements of the lattice ℤ_{θ,ϵ}(τ) (C.16). Choose f in the form f = a − b − c to obtain

e_{θ}(a × b + 2b × c) (φ_{θ,ϵ}(c, z − w|ℏ)φ_{θ,ϵ}(a − c, z|ℏ)φ_{θ,ϵ}(b + c, w|ℏ)
−φ_{θ,ϵ}(a − b − c, z − w|ℏ)φ_{θ,ϵ}(a − c, w|ℏ)φ_{θ,ϵ}(b + c, z|ℏ)) = 0 .

This equality can be transform to the form

φ_{θ,ϵ}(a + 2ℏ, z|ℏ)φ_{θ,ϵ}(b, w|ℏ)e_{θ}(a × b + 2b × c)
× (ζ_{θ,ϵ}(c + ℏ) − ζ_{θ,ϵ}(a − b − c + ℏ) + ζ_{θ,ϵ}(a − c + ℏ) − ζ_{θ,ϵ}(b + c + ℏ)) = 0

by means of (A.27). If b = 0 then the l.h.s. vanishes and we come to (6.1). Let b ≠ 0 and consider the shift c → c + jb, where j ∈ ℤ. The shift does not change the exponential factor. Take the sum over the orbit generated by the shifts

∑_{j∈ℤ}ζ_{θ,ϵ}(c + jb + ℏ) − ζ_{θ,ϵ}(a − b − jb − c + ℏ) + ζ_{θ,ϵ}(a − c + jb + ℏ) − ζ_{θ,ϵ}(b + c + jb + ℏ) .

One can see that the neighboring terms in the series vanish and we come to (6.1) for an arbitrary b. □

This Proposition allows us to define the associative algebra P̂_{θ,ϵ,τ,ℏ} by the relation

R(z − w)L_{1}^{ℏ}(z)L_{2}^{ℏ}(w) = L_{2}^{ℏ}(w)L_{1}^{ℏ}(z)R(z − w) ,  (6.2)

where

L^{ℏ}(z) = Ŝ_{0}φ_{θϵ}(0, z|ℏ)T_{0} + ∑_{α} Ŝ_{−α}φ_{θ,ϵ}(α, z|ℏ)T_{α} .

In order to obtain the relations in P̂_{θ,ϵ,τ,ℏ} one should exclude the spectral parameters z, w from (6.2).

Lemma 6.1 The relations in the associative algebra P̂_{θ,τ,ℏ} assume the form

∑_{c} Ŝ_{a−c}Ŝ_{b+c}e_{θ}(c · (a − b))f_{θ,ϵ}(a, b, c|ℏ) = 0 , for any a, b ∈ ℤ ⊕ ℤ ,  (6.3)

where

f_{θ,ϵ}(a, b, c|ℏ) = ζ_{θ,ϵ}(c + ℏ) − ζ_{θ,ϵ}(a − b − c + ℏ) − ζ_{θ,ϵ}(b + c + ℏ) + ζ_{θ,ϵ}(a − c + ℏ)  (6.4)

= −\frac{ϑ′(0)ϑ_{θ,ϵ}(α + 2ℏ)ϑ_{θ,ϵ}(β)ϑ_{θ,ϵ}(β − α + 2γ)}{ϑ_{θ,ϵ}(α − γ + ℏ)ϑ_{θ,ϵ}(β + γ + ℏ)ϑ_{θ,ϵ}(β − α + γ + ℏ)ϑ_{θ,ϵ}(γ + ℏ)} .

and ϑ_{θ,ϵ}, ζ_{θ,ϵ}(.) are the constants (C.17), (C.18).

Proof. Consider in (6.2) the matrix element T_{a} ⊗ T_{b}. We come to the relation

∑_{c} Ŝ_{a−c}Ŝ_{b+c}{φ_{θ,ϵ}(c, z − w|ℏ)φ_{θ,ϵ}(a − c, z|ℏ)φ_{θ,ϵ}(b + c, w|ℏ)  (6.5)

−φ_{θ,ϵ}(a − b − c, z − w|ℏ)φ_{θ,ϵ}(a − c, w|ℏ)φ_{θ}(b + c, z|ℏ)}e_{θ}(c · (a − b)) = 0 .

The expression in the brackets { } is equal to

e_{θ}((a + 2ℏ)z + (b + ℏ)w){ϕ_{θ,ϵ}(c, z − w|ℏ)ϕ_{θ,ϵ}(a − c, z|ℏ)ϕ_{θ,ϵ}(b + c, z − w|ℏ)
−ϕ_{θ,ϵ}(a − b − c, z − w|ℏ)ϕ_{θ,ϵ}(a − c, w|ℏ)ϕ_{θ,ϵ}(b + c, z|ℏ)} ,

where ϕ_{θ,ϵ}(c, z|ℏ) = ϕ((ϵ_{1}c_{1} + τϵ_{2}c_{2})θ + ℏ, z). Finally, by using (A.27) we come from (6.5) to (6.4). □

We rewrite (6.3) in the form most resembling the original Sklyanin relations [14]. Let 𝔄(α, β) (respectively, 𝔖(α, β)) be the operator of antisymmetrization (respectively, symmetrization) with respect to the permutations of two indices (α, β).

Proposition 6.2 The relations (6.4) are equivalent to the commutator relations

[Ŝ_{0}, Ŝ_{b}] = − ∑_{γ≠b} Ŝ_{−γ}Ŝ_{b+γ}e_{θ}(b · γ)\frac{f_{θ,ϵ}(0, b, γ|ℏ)}{f_{θ,ϵ}(0, b, 0|ℏ)} ,  (6.6)

[Ŝ_{α}, Ŝ_{β}] = 𝔄(α, β) \{(Ŝ_{α+β}Ŝ_{0}e_{θ}(−β · α) − Ŝ_{0}Ŝ_{α+β}e_{θ,ϵ}(β · α)) \frac{f_{θ,ϵ}(α, β, α|ℏ)}{f_{θ,ϵ}(α, β, 0|ℏ)}  (6.7)

+ ∑_{γ≠α,−β} (Ŝ_{α−γ}Ŝ_{β+γ}e_{θ}(γ · (α − β)) − Ŝ_{β+γ}Ŝ_{α−γ}e_{θ}(−γ · (α − β))) \frac{f_{θ,ϵ}(α, β, γ|ℏ)}{f_{θ,ϵ}(α, β, 0|ℏ)}\} ,

where C_{θ}(α, β) are sl(N, ℂ) structure constants (B.8), and

𝔖(α, β) \{(Ŝ_{α+β}Ŝ_{0}e_{θ}(−β · α) − Ŝ_{0}Ŝ_{α+β}e_{θ,ϵ}(β · α)) \frac{f_{θ,ϵ}(α, β, α|ℏ)}{f_{θ,ϵ}(α, β, 0|ℏ)}

+ ∑_{γ≠α,−β} (Ŝ_{α−γ}Ŝ_{β+γ}e_{θ}(γ · (α − β)) − Ŝ_{β+γ}Ŝ_{α−γ}e_{θ}(−γ · (α − β))) \frac{f_{θ,ϵ}(α, β, γ|ℏ)}{f_{θ,ϵ}(α, β, 0|ℏ)}\} = 0 .

Proof. The relation (6.3) can be rewritten in the form

∑_{c} (Ŝ_{a−c}Ŝ_{b+c}e_{θ}(c · (a − b)) − Ŝ_{b+c}Ŝ_{a−c}e_{θ}(c · (a − b))) f_{θ,ϵ}(a, b, c|ℏ) = 0 .

due to the equality f_{θ,ϵ}(a, b, c|ℏ) = −f_{θ,ϵ}(a, b, a−b−c|ℏ). For the case a = 0 this relation assumes the form (6.6). To come to (6.7) we single out the two terms with c = 0 and put them in the left hand side. In the right hand side we first write down the terms with c = a and c = −b and obtain the required relations. □

It would be interesting to describe a quantum version of the whole bihamiltonian structure for elliptic rotators on sin^{∗} or Ham(T^{2}). Posing the problem more generally, the bihamiltonian recursion precedure of generating the conserved quantities from a Casimir function for a linear family of Poisson structures (see [19]) might have a quantum analog as an expansion of a central element for a linear family of associative algebras. The latter seems to be a very strong requirement on a pair of associative algebras and it would be very interesting to find any non-trivial example of this kind. Namely, one is looking for a pair of associative algebras, such that their mixed associator satisfies some consistency condition. This would provide the most straightforward quantization for a system with a bihamiltonian structure.

Acknowledgments

We are grateful for hospitality to the Max-Planck-Institut fur Mathematik in Bonn, where most of this work was done. B.K. is also grateful for hospitality to the University of Nice and IHES in Bures-sur-Yvette. The work of B.K. was partially supported by an NSERC research grant. The work of A.L. and M.O. was supported in part by grants RFBR-03-02-17554, NSch-1999-2003.2 and CRDF 12729.

7 Appendix

7.1 Appendix A. Elliptic functions.

Here we summarize the main formulae for elliptic functions. Consider an elliptic curve

E_{τ} = ℂ/(ℤ + τℤ) , q = e(τ) = exp(2πiτ) .  (A.1)

The basic element in our consideration is the theta function:

ϑ(z; τ) = q^{\frac{1}{8}} ∑_{n∈Z}(−1)^{n}e^{πi(n(n+1)τ+2nz)} =  (A.2)

q^{\frac{1}{8}} e^{− \frac{iπ}{4}} (e^{iπz} − e^{−iπz}) ∏_{n=1}^{∞}(1 − q^{n})(1 − q^{n}e^{2iπz})(1 − q^{n}e^{−2iπz}) .

1. The Weierstrass functions:

σ(z; τ) = exp(ηz^{2})\frac{ϑ(z; τ)}{ϑ′(0; τ)} ,  (A.3)

where

η(τ) = −\frac{1}{6} \frac{ϑ′′′(0; τ)}{ϑ′(0; τ)} .  (A.4)
ζ(z; τ) = ∂_{z} log ϑ(z; τ) + 2η(τ)z , ζ(z; τ) ∼ \frac{1}{z} + O(z^{3}) .  (A.5)
℘(z; τ) = −∂_{z}ζ(z; τ) .  (A.6)

℘(u; τ) = \frac{1}{u^{2}} + + ∑_{j,k}^{′} (\frac{1}{(j + kτ + u)^{2}} − \frac{1}{(j + kτ)^{2}} ) .  (A.7)

2. Function ϕ:

ϕ(u, z) = \frac{ϑ(u + z)ϑ′(0)}{ϑ(u)ϑ(z)}  (A.8)

It has a pole at z = 0 and the expansion

ϕ(u, z) = \frac{1}{z} + ζ(u|τ) + 2η(τ)z + \frac{z}{2}((ζ(u; τ) + 2η(τ)z)^{2} − ℘(u)) + …,  (A.9)

as z → 0. Furthermore,

ϕ(u, z)^{−1}∂_{u}ϕ(u, z) = ζ(u + z) − ζ(u) + 2η(τ)z .  (A.10)
ϕ(u, z) = exp(−2ηuz)\frac{σ(u + z)}{σ(u)σ(z)} ,  (A.11)
ϕ(u, z)ϕ(−u, z) = ℘(z) − ℘(u) .  (A.12)

3. Quasi-periodicity:

ϑ(z + 1; τ) = −ϑ(z; τ) , ϑ(z + τ; τ) = −e(−\frac{1}{2}τ − z)ϑ(z; τ) .  (A.13)
ζ(z + 1; τ) = ζ(z; τ) − 2η , ζ(z + τ; τ) = ζ(z; τ) − 2(πi + ητ) .  (A.14)
℘(u + 1; τ) = ℘(u + τ; τ) = ℘(u; τ) .  (A.15)
ϕ(u + 1, z) = ϕ(u, z) , ϕ(u + τ, z) = e(−z)ϕ(u, z) .  (A.16)

4. Parity:

ϑ(−z; τ) = −ϑ(z; τ) .  (A.17)
ζ(−z; τ) = −ζ(z; τ) .  (A.18)
℘(−u; τ) = ℘(u; τ) .  (A.19)
ϕ(−u, −z) = −ϕ(u, z) .  (A.20)

5. The Fay three-section formula:

ϕ(u_{1}, z_{1})ϕ(u_{2}, z_{2}) − ϕ(u_{1} + u_{2}, z_{1})ϕ(u_{2}, z_{2} − z_{1}) − ϕ(u_{1} + u_{2}, z_{2})ϕ(u_{1}, z_{1} − z_{2}) = 0 . (A.21)

Particular cases of this formula are (A.12), the Calogero functional equation

ϕ(u, z)∂_{v}ϕ(v, z) − ϕ(v, z)∂_{u}ϕ(u, z) = (℘(v) − ℘(u))ϕ(u + v, z) ,  (A.22)

and

ϕ(u_{1}, z)ϕ(u_{2}, z) − ϕ(u_{1} + u_{2}, z)(ζ(u_{1}) + ζ(u_{2}) − 2η(τ)(u_{1} + u_{2})) + ∂_{z}ϕ(u_{1} + u_{2}, z) = 0 . (A.23)

For u_{1} + u_{2} + u_{3} = 0,

ϕ(u_{1}, z)ϕ(u_{2}, z)ϕ(u_{3}, z) = [℘(z) − ℘(u_{3})] [ζ(u_{1}) + ζ(u_{2}) + ζ(u_{3} − z) + ζ(z)] ,  (A.24)

as follows from (A.10), (A.12),and (A.23). Then

ϕ(u_{1}, z)ϕ(u_{2}, z)ϕ(u_{3}, z)|_{z→0} = \frac{1}{z^{3}} + \frac{1}{z^{2}} [ζ(u_{1}) + ζ(u_{2}) + ζ(u_{3})]  (A.25)

−\frac{1}{2}℘′(u_{3}) − ℘(u_{3}) [ζ(u_{1}) + ζ(u_{2}) + ζ(u_{3})] + O(z) .

¿From (A.21) and (A.10) we have

ϕ(u_{1}, z)ϕ(u − u_{1}, z) = ϕ(u, z)(ζ(u_{1}) + ζ(u − u_{1}) − ζ(u + z) + ζ(z)) .  (A.26)

Another important relation is

ϕ(v, z − w)ϕ(u_{1} − v, z)ϕ(u_{2} + v, w) − ϕ(u_{1} − u_{2} − v, z − w)ϕ(u_{2} + v, z)ϕ(u_{1} − v, w) = (A.27)
ϕ(u_{1}, z)ϕ(u_{2}, w)f(u_{1}, u_{2}, v) ,

where

f(u_{1}, u_{2}, v) = ζ(v) − ζ(u_{1} − u_{2} − v) + ζ(u_{1} − v) − ζ(u_{2} + v) .  (A.28)

One can rewrite the last function as

f(u_{1}, u_{2}, v) = −\frac{ϑ′(0)ϑ(u_{1})ϑ(u_{2})ϑ(u_{2} − u_{1} + 2v)}{ϑ(u_{1} − v)ϑ(u_{2} + v)ϑ(u_{2} − u_{1} + v)ϑ(v)} .  (A.29)

To prove (A.27) one should consider the expression

\frac{ϕ(v, z − w)ϕ(u_{1} − v, z)ϕ(u_{2} + v, w) − ϕ(u_{1} − u_{2} − v, z − w)ϕ(u_{2} + v, z)ϕ(u_{1} − v, w)}{ϕ(u_{1}, z)ϕ(u_{2}, w)} .

It is a doubly periodic function in z and w without poles. Therefore, it is a constant, which depends on u_{1}, u_{2}, v. This constant is equal to f(u_{1}, u_{2}, v) (A.28). It is easy to check that the elliptic functions (A.28) and (A.29) coincide.

A particular case of (A.27), which corresponds to the case v = u_{1} (or v = −u_{2}), is the Fay identity (A.21). Another particular case comes from u_{2} = 0 (or u_{1} = 0):

ϕ(v, z − w)ϕ(u − v, z)ϕ(v, w) − ϕ(u − v, z − w)ϕ(v, z)ϕ(u − v, w) =  (A.30)
ϕ(u_{1}, z)(℘(v) − ℘(u − v)) .

7.2 Appendix B. Elliptic constants related to gl(N, ℂ).

Let 1 ≤ k < N be a coprime number with respect to N and set θ = \frac{k}{N}. Define

e_{θ}(z) = exp(2πiθz) ,  (B.1)
Q = diag(e_{θ}(1), …, e_{θ}(m), …, 1)  (B.2)

Λ = ( \begin{matrix}0 & 1 & 0 & ⋯ & 0 \\ 0 & 0 & 1 & ⋯ & 0 \\ ⋮ & ⋮ & ... & ... & ⋮ \\ 0 & 0 & 0 & ⋯ & 1 \\ 1 & 0 & 0 & ⋯ & 0 \end{matrix} ) .  (B.3)

Consider a finite two-dimensional lattice of order N^{2}

ℤ_{N}^{(2)} = ℤ/Nℤ ⊕ ℤ/Nℤ .

The matrices

T_{a} = \frac{1}{2πiθ} e_{θ}(\frac{a_{1}a_{2}}{2} )Q^{a_{1}}Λ^{a_{2}} , (a = (a_{1}, a_{2}) ∈ ℤ^{(2)}  (B.4)

generate the basis in gl(N, ℂ). We use the Greek letters for the elements of the lattice

ℤ̃_{N}^{(2)} = ℤ_{N}^{(2)} \ (0, 0) .  (B.5)

Thus, {T_{α}} define a basis in sl(N, ℂ). Since

T_{a}T_{b} = \frac{1}{2πθ} e_{θ}(−\frac{a × b}{2})T_{a+b} , (a × b = a_{1}b_{2} − a_{2}b_{1})  (B.6)

the commutation relations in this basis assume the form

[T_{α}, T_{β}] = C_{θ}(α, β)T_{α+β} ,  (B.7)

where

C_{θ}(α, β) = \frac{1}{πθ} sin πθ(α × β) .  (B.8)

Let

ℤ_{θ}^{(2)}(τ) = (γ_{1} + γ_{2}τ)θ , γ ∈ ℤ̃_{N}^{(2)}  (B.9)

be a regular lattice of order N^{2} − 1 on E_{τ}. Introduce the following constants on ℤ_{θ}^{(2)}(τ):

ϑ_{θ}(γ) = ϑ((γ_{1} + γ_{2}τ)θ) ,  (B.10)
ζ_{θ}(γ) = ζ((γ_{1} + γ_{2}τ)θ) , ℘_{θ}(γ) = ℘((γ_{1} + γ_{2}τ)θ) ,  (B.11)

and the quasi-periodic functions on E_{τ}

ϕ_{θ}(γ, z) = ϕ((γ_{1} + γ_{2}τ)θ, z) ,  (B.12)
φ_{θ}(γ, z) = e_{θ}(γ_{2}z)ϕ_{θ}(γ, z) ,  (B.13)
f_{θ}(γ, z) = e_{θ}(γ_{2}z)∂_{u}ϕ(u, z)|_{u=(γ_{1}+γ_{2}τ)θ} .  (B.14)

It follows from (A.8) that

φ_{θ}(γ, z + 1) = e_{θ}(γ_{2})φ_{θ}(γ, z) , φ_{θ}(γ, z + τ)e_{θ}(−γ_{1})φ_{θ}(γ, z) .  (B.15)

7.3 Appendix C. Non-commutative torus

1. Definition and representation. The non-commutative torus A_{θ} is a unital algebra with the two generators (U_{1}, U_{2}) that satisfy the relation

U_{1}U_{2} = e_{θ}^{−1}U_{2}U_{1}, e_{θ} = e^{2πiθ}, θ ∈ [0, 1) .  (C.1)

Elements of A_{θ} are the double sums

A_{θ} = {X = ∑_{a_{1},a_{2}∈ℤ}c_{a_{1},a_{2}}U_{1}^{a_{1}}U_{2}^{a_{2}} | c_{a_{1},a_{2}} ∈ ℂ} ,

where c_{a_{1},a_{2}} are elements of the ring 𝔖 of rapidly decreasing sequences on ℤ^{2}

𝔖 = {c_{a_{1},a_{2}} | sup_{a_{1},a_{2}∈ℤ}(1 + a_{1}^{2} + a_{2}^{2})^{k}|c_{a_{1},a_{2}}|^{2} < ∞ forall k ∈ ℕ} .  (C.2)

The trace functional tr(x) on A_{θ} is defined by

tr(X) = c_{00} .  (C.3)

It satisfies the evident identities

tr(1) = 1, tr(XY) = tr(Y X) .

The dual space to 𝔖 is the space

𝔖′ = {S_{a_{1},a_{2}} | ∑_{a_{1},a_{2}} c_{a_{1},a_{2}}S_{−a_{1},−a_{2}} < ∞, c_{a_{1},a_{2}} ∈ 𝔖} .  (C.4)

The relation with the commutative algebra of smooth functions on the two-dimensional torus

T^{2} = {ℝ^{2}/ℤ ⊕ ℤ} ∼ {0 < x_{1} ≤ 1, 0 < x_{2} ≤ 1}.  (C.5)

comes from the identification

U_{1} → e(x), U_{2} → e(y),  (C.6)

while the multiplication on T^{2} becomes the Moyal multiplication:

(f ∗ g)(x) := fg + ∑_{n=1}^{∞} \frac{(πθ)^{n}}{n!} ε_{r_{1}s_{1}} … ε_{r_{n}s_{n}}(∂_{r_{1}...r_{n}}^{n}f)(∂_{s_{1}...s_{n}}^{n}g).  (C.7)

The trace functional (C.3) in the Moyal identification is the integral

trf = −\frac{1}{4π^{2}} ∫_{A_{θ}} fdx_{1}dx_{2} = f_{00} .  (C.8)

We can identify U_{1}, U_{2} with matrices from GL(∞). Define GL(∞) as the associative algebra of infinite matrices c_{jk}E_{jk}, where E_{jk} = ||δ_{jk}||, such that

sup_{j,k∈ℤ}|c_{jk}|^{2}|j − k|^{n} < ∞ for n ∈ ℕ .

Consider the following two matrices from GL(∞):

Q = diag(e(jθ)) and Λ = ||δ_{j,j+1}|| , j ∈ ℤ .

We have the following identification

U_{1} → Q, U_{2} → Λ .  (C.9)

Another useful realization of A_{θ} in the Schwartz space on ℝ by the operators

U_{1}f(x) = f(x − θ) , U_{2}f(x) = exp(2πix)f(x) .  (C.10)

2. sin-algebra. Define the following quadratic combinations of the generators

T_{α} = \frac{i}{2πθ} e (\frac{α_{1}α_{2}}{2} θ) U_{1}^{m}U_{2}^{n} α ∈ ℤ̃^{(2)} ,  (C.11)

ℤ̃^{(2)} = {α = (α_{1}, α_{2}) , α_{j} ∈ ℤ, α ≠ (0, 0)} .

Their commutator has the form of the sin-algebra

[T_{α}, T_{β}] = C_{θ}(α, β)T_{α+β} ,  (C.12)

where

C_{θ}(α, β) = \frac{1}{πθ} sin πθ(α × β) .  (C.13)

We denote by sin_{θ} the Lie algebra with the generators T_{α} over the ring 𝔖 (C.2)

ψ = ∑_{α} ψ_{α}T_{α}, ψ_{α} ∈ 𝔖 ,  (C.14)

and by SIN_{θ} the group of invertible elements from A_{θ}. The coalgebra sin_{θ}^{∗} is the linear space

sin_{θ}^{∗} = {S = ∑_{α∈ℤ̃^{(2)}} s_{α}T α, s_{α} ∈ 𝔖′} .

In the Moyal representation (C.7) the commutator of sin_{θ} has the form

[f(x, y), g(x, y)] = \{f, g\}^{θ} := \frac{1}{θ}(f ∗ g − g ∗ f)  (C.15)

3. Elliptic constants related to NCT A_{θ} Introduce two numbers ϵ = (ϵ_{1}, ϵ_{2}) such that ϵ_{a}θ < 1 and ϵ_{a}θ are irrational. Consider the dense set ℤ_{θ,ϵ}(τ) in E_{τ}:

ℤ_{θ,ϵ}(τ) = {(ϵ_{1}γ_{1} + τϵ_{2}γ_{2})θ ∈ E_{τ} | (γ_{1}, γ_{2}) ∈ ℤ̃^{(2)}} .  (C.16)

The corresponding elliptic functions with the arguments from ℤ_{θ,ϵ}(τ) are as follows:

ϑ_{θ,ϵ}(γ) = ϑ((ϵ_{1}γ_{1} + ϵ_{2}γ_{2}τ)θ) ,  (C.17)
ζ_{θ,ϵ}(γ) = ζ((ϵ_{1}γ_{1} + τϵ_{2}γ_{2})θ) , ℘_{θ,ϵ}(γ) = ℘((ϵ_{1}γ_{1} + τϵ_{2}γ_{2})θ) .  (C.18)
ϕ_{θ,ϵ}(γ, z) = ϕ(−(ϵ_{1}γ_{1} + τϵ_{2}γ_{2})θ, z) ,  (C.19)
φ_{θ,ϵ}(γ) = e_{θ}(ϵ_{2}γ_{2}z)ϕ_{θ,ϵ}(γ, z) .  (C.20)
f_{θ,ϵ}(γ) = e_{θ}(ϵ_{2}γ_{2}z)∂_{u}ϕ(u, z)|_{u=(ϵ_{1}γ_{1}+τϵ_{2}γ_{2})θ} .  (C.21)

4. Dispersionless limit.

In the limit θ → 0 the Lie algebra sin_{θ} becomes the Lie algebra of Hamiltonian functions

Ham(T^{2}) ∼ C^{∞}(T^{2})/ℂ  (C.22)

equipped with the canonical Poisson brackets. In Ham(T^{2}) we have the Fourier basis

e(α · x) = exp(2πi(α_{1}x_{1} + α_{2}x_{2}))  (C.23)

instead of the basis (C.11). The commutator (C.12) becomes

[e(αx), e(βx)] = (α × β)e((α + β) · x) .

The algebra Ham(T^{2}) (without constant Hamiltonians) is isomorphic to Lie algebra SV ect_{0}(T^{2}) of the divergence-free zero-flux vector fields on T^{2} equipped with the area form −4π^{2}dx_{1}dx_{2}. Let h(x_{1}, x_{2}) ∈ Ham(T^{2}). Then the Hamiltonian field V_{h} corresponding to the Hamiltonian function h is

V_{h} = −\frac{1}{4π^{2}} ((∂_{2}h)∂_{1} − (∂_{1}h)∂_{2}) ,  (C.24)

while

[V_{h}, V_{h′}] = V_{{h,h′}} .  (C.25)

For f(x) = ∑_{α} f_{α}e(α · x)

∫_{T^{2}} f = −\frac{1}{4π^{2}} f_{0} .  (C.26)

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