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Advances in Mathematical Physics

Volume 2011 (2011), Article ID 252186, 18 pages

http://dx.doi.org/10.1155/2011/252186

## Integer Solutions of Integral Inequalities and -Invariant Jacobian Poisson Structures

^{1}Dipartimento di Matematica Pura e Applicazioni, Università degli Milano Bicocca, Via R.Cozzi 53, 20125 Milano, Italy^{2}Département de Mathématiques 2, Laboratoire Angevin de Recherche en Mathématiques Université D'Angers, boulevard Lavoisier, 49045 Angers, France^{3}Mathematics Research Unit at Luxembourg, University of Luxembourg, 6 rue Richard Coudenhove-Kalergi, 1359 Luxembourg City, Luxembourg

Received 14 April 2011; Accepted 3 June 2011

Academic Editor: Yao-Zhong Zhang

Copyright © 2011 G. Ortenzi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We study the Jacobian Poisson structures in any dimension invariant with respect to the discrete Heisenberg group. The classification problem is related to the discrete volume of suitable solids. Particular attention is given to dimension 3 whose simplest example is the Artin-Schelter-Tate Poisson tensors.

#### 1. Introduction

This paper continues the authors' program of studies of the Heisenberg invariance properties of polynomial Poisson algebras which were started in [1] and extended in [2, 3]. Formally speaking, we consider the polynomials in variables over and the action of some subgroup of generated by the shifts operators and by the operators , where . We are interested in the polynomial Poisson brackets on which are “stable” under this actions (we will give more precise definition below).

The most famous examples of the Heisenberg invariant polynomial Poisson structures are the Sklyanin-Odesskii-Feigin-Artin-Tate quadratic Poisson brackets known also as the elliptic Poisson structures. One can also think about these algebras like the “quasiclassical limits” of elliptic Sklyanin associative algebras. These is a class of Noetherian graded associative algebras which are Koszul, Cohen-Macaulay, and have the same Hilbert function as a polynomial ring with variables. The above-mentioned Heisenberg group action provides the automorphisms of Sklyanin algebras which are compatible with the grading and defines an -action on the elliptic quadratic Poisson structures on . The latter are identified with Poisson structures on some moduli spaces of the degree and rank vector bundles with parabolic structure (= the flag on the elliptic curve ). We will denote this elliptic Poisson algebras by . The algebras arise in the Feigin-Odesskii “deformational” approach and form a subclass of polynomial Poisson structures. A comprehensive review of elliptic algebras can be found in [4] to which we refer for all additional information. We will mention only that as we have proved in [3] all elliptic Poisson algebras (being in particular Heisenberg-invariant) are unimodular.

Another interesting class of polynomial Poisson structures consists of so-called Jacobian Poisson structures (JPS). These structures are a special case of Nambu-Poisson structures. Their rank is two, and the Jacobian Poisson bracket of two polynomials and is given by the determinant of Jacobi matrix of functions . The polynomials , are Casimirs of the bracket and under some mild condition of independence are generators of the centrum for the Jacobian Poisson algebra structure on . This type of Poisson algebras was intensively studied (due to their natural origin and relative simplicity) in a huge number of publications among which we should mention [1, 5–9].

There are some beautiful intersections between two described types of polynomial Poisson structures: when we are restricting ourselves to the class of quadratic Poisson brackets then there are only Artin-Schelter-Tate () and Sklyanin () algebras which are both elliptic and Jacobian. It is no longer true for . The relations between the Sklyanin Poisson algebras whose centrum has dimension 1 (for odd) and 2 (for even) in the case and is generated by Casimirs for for are in general quite obscure. We can easily found that sometimes the JPS structures correspond to some degenerations of the Sklyanin elliptic algebras. One example of such JPS for was remarked in [8] and was attributed to so-called Briesckorn-Pham polynomials for It is easy to check that the homogeneous quintic (see Section 4.2) defines a Casimir for some rational degeneration of (one of) elliptic algebras and if it satisfies the -invariance condition.

In this paper, we will study the Jacobian Poisson structures in any number of variables which are Heisenberg-invariant and we relate all such structures to some graded subvector space of polynomial algebra. This vector space is completely determined by some enumerative problem of a number-theoretic type. More precisely, the homogeneous subspace of of degree is in bijection with integer solutions of a system of Diophant inequalities. Geometric interpretation of the dimension of is described in terms of integer points in a convex polytope given by this Diophant system. In the special case of dimension 3, is a subalgebra of polynomial algebra with 3 variables and all JPS are given by this space. We solve explicitly the enumerative problem in this case and obtain a complete classification of the -invariant not necessarily quadratic Jacobian Poisson algebras with three generators. As a byproduct, we explicitly compute the Poincaré series of . In this dimension, we observe that the -invariant JPS of degree 5 is given by the Casimir sextic . This structure is a “projectively dual” to the Artin-Schelter-Tate elliptic Poisson structure which is the -invariant JPS given by the cubic where . In fact, the algebraic variety is the (generically) projectively dual to the elliptic curve .

The paper is organized as follows: in Section 2, we remind a definition of the Heisenberg group in the Schroedinger representation and describe its action on Poisson polynomial tensors and also the definition of JPS. In Section 3, we treat the above mentioned enumerative problem in dimension 3. The last section concerns the case of any dimension. Here, we discuss some possible approaches to the general enumerative question.

#### 2. Preliminary Facts

Throughout of this paper, is a field of characteristic zero. Let us start by remembering some elementary notions of the Poisson geometry.

##### 2.1. Poisson Algebras and Poisson Manifold

Let be a commutative -algebra. One says that is a Poisson algebra if is endowed with a Lie bracket, indicated with , which is also a biderivation. One can also say that is endowed with a Poisson structure, and therefore, the bracket is called the Poisson bracket. Elements of the center are called Casimirs: is a Casimir if for all .

A Poisson manifold (smooth, algebraic, etc.) is a manifold whose function algebra (, regular, etc.) is endowed with a Poisson bracket.

As examples of Poisson structures let us consider a particular subclass of Poisson structures which are uniquely characterized by their Casimirs. In the dimension 4, let be two elements of , where .

On , a Poisson structure is defined by or more explicitly (mod ) Sklyanin had introduced this Poisson algebra which carries today his name in a Hamiltonian approach to the continuous and discrete integrable Landau-Lifshitz models [10, 11]. He showed that the Hamiltonian structure of the classical model is completely determined by two quadratic “Casimirs”. The Sklyanin Poisson algebra is also called elliptic due to its relations with an elliptic curve. The elliptic curve enters in the game from the geometric side. The symplectic foliation of Sklyanin's structure is too complicated. This is because the structure is degenerated and looks quite different from a symplectic one. But the intersection locus of two Casimirs in the affine space of dimension four (one can consider also the projective situation) is an elliptic curve given by two quadrics . We can think about this curve as a complete intersection of the couple , embedded in (as it was observed in Sklyanin's initial paper).

A possible generalization one can be obtained considering polynomials in with coordinates , . We can define a bilinear differential operation by This operation, which gives a Poisson algebra structure on , is called a Jacobian Poisson structure (JPS), and it is a partial case of more general -ary Nambu operation given by an antisymmetric -polyvector field introduced by Nambu [6] and was extensively studied by Takhtajan [5].

The polynomials , are Casimir functions for the brackets (2.5).

There exists a second generalization of the Sklyanin algebra that we will describe briefly in the next subsection (see, for details, [4]).

##### 2.2. Elliptic Poisson Algebras and

(We report here this subsection from [2] for sake of self-consistency).

These algebras, defined by Feĭgin and Odesskiĭ, arise as quasiclassical limits of elliptic associative algebras and [12, 13].

Let be an integral lattice generated by 1 and , with . Consider the elliptic curve and a point on this curve.

In their article [13], given , mutually prime, Odesskiĭ and Feĭgin construct an algebra, called elliptic, , as an algebra defined by generators and the following relations: where are theta functions [13].

These family of algebras has the following properties: (1)the center of the algebra , for generic and , is the algebra of polynomial of variables of degree ,(2) is commutative, (3) is commutative for all , (4), if (mod ), (5)the maps et , where , define automorphisms of the algebra ,(6)the algebras are deformations of polynomial algebras. The associated Poisson structure is denoted by , (7)among the algebras , only (the Artin-Schelter-Tate algebra) and the Sklyanin algebra are Jacobian Poisson structures.

##### 2.3. The Heisenberg Invariant Poisson Structures

###### 2.3.1. The G-Invariant Poisson Structures

Let be a group acting on a Poisson algebra .

*Definition 2.1. *A Poisson bracket on is said to be a -invariant if acts on by Poisson automorphisms.

In other words, for every , the morphism , is an automorphism and the following diagram is a commutative:

(2.7)

###### 2.3.2. The H-Invariant Poisson Structures

In their paper [2], the authors introduced the notion of -invariant Poisson structures. That is, a special case of a -invariant structure when in the finite Heisenberg group and is the polynomial algebra. Let us remember this notion.

Let be a complex vector space of dimension and a basis of . Take the th primitive root of unity .

Consider of defined by The Heisenberg of dimension is nothing else that the subspace generated by and .

From now on, we assume that , with as a basis and consider the coordinate ring .

Naturally, and act by automorphisms on the algebra as follows: We introduced in [2] the notion of -degree on the polynomial algebra ]. The -degree of a monomial is the positive integer if and if not. The -degree of is denoted . A -degree of a polynomial is the highest -degree of its monomials.

For simplicity, the -invariance condition will be referred from now on just as -invariance. An -invariant Poisson bracket on is nothing but a bracket on which satisfy the following: for all .

The invariance is, in some sense, a “discrete” homogeneity.

Proposition 2.2 (see [2]). *The Sklyanin-Odesskii-Feigin Poisson algebras are -invariant Poisson algebras.*

Therefore, an -invariant Poisson structures on the polynomial algebra includes as the Sklyanin Poisson algebra or more generally of the Odesskii-Feigin Poisson algebras.

In this paper, we are interested in the intersection of the two classes of generalizations of Artin-Shelter-Tate-Sklyanin Poisson algebras: JPS and -invariant Poisson structures.

Proposition 2.3 (see [2]). *If is an -invariant polynomial Poisson bracket, the usual polynomial degree of the monomial of equals to , .*

Proposition 2.4 (see [2]). *Let .**For all ,
*

*3. -Invariant JPS in Dimension 3*

*We consider first a generalization of Artin-Schelter-Tate quadratic Poisson algebras. Let be the polynomial algebra with 3 generators. For every , we have a JPS on given by
where is a cyclic permutation of . Let be the set of all such that is an -invariant Poisson structure.*

*Proposition 3.1. If is a homogeneous polynomial, then and .*

*Proof. *Let be a cyclic permutation of . One has
Using Proposition 2.4, we conclude that for all , .

It gives that .

On the other hand, from (3.2), one has . And we get the second half of the proposition.

*Proposition 3.2. is a subalgebra of .*

*Proof. *Let . It is clear that for all , belongs to .

Let us denote by the JPS associated with the polynomial . It is easy to verify that . Therefore, it is clear that the -invariance condition is verified for the JPS associated to the polynomial .

*We endow with the usual grading of the polynomial algebra . For , an element of , we denote by its usual weight degree. We denote by the homogeneous subspace of of degree .*

*Proposition 3.3. If 3 does not divide (in other words ), then .*

*Proof. *First of all . We suppose now that . Let . Then, . It follows from Proposition 2.3 and the definition of the Poisson brackets that there exists such that . The result follows from the fact that .

*Set , where . We suppose that . We want to find all such that and, therefore, the dimension as -vector space.*

*Proposition 3.4. There exist , , and such that
*

*Proof. *This is a direct consequence of Proposition 3.3.

*Proposition 3.5. The system equation (3.4) has as solutions the following set:
where , and live in the polygon given by the following inequalities in *

*Remark 3.6. *For , one obtains the Artin-Schelter-Tate Poisson algebra which is the JPS given by the Casimir , . Suppose that , then it can take the form
where . The interesting feature of this Poisson algebra is that their polynomial character is preserved even after the following nonalgebraic changes of variables. Let
The polynomial in the coordinates has the form
The Poisson bracket is also polynomial (which is not evident at all!) and has the same form
where is the cyclic permutation of . This JPS structure is no longer satisfying the Heisenberg invariance condition. But it is invariant with respect the following toric action: given by

Put . Then, the polynomial is also homogeneous in and defines an elliptic curve in the weighted projective space .

The similar change of variables
defines the JPS structure invariant with respect to the torus action given by
and related to the elliptic curve in the weighted projective space .

These structures had appeared in [1], their Poisson cohomology was studied by Pichereau [14], and their relation to the noncommutative del Pezzo surfaces and Calabi-Yau algebras were discussed in [15].

*Proposition 3.7. The subset of given by the system (3.6) is a triangle with , , and as vertices. Then, *

*Remark 3.8. *For , the case of Figure 1, the generic Heisenberg-invariant JPS is given by the sextic polynomial
The corresponding Poisson bracket takes the form
where are the cyclic permutations of .

This new JPS should be considered as the “projectively dual” to the Artin-Schelter-Tate JPS, since the algebraic variety is generically the projective dual curve in to the elliptic curve

To establish the exact duality and the explicit values of the coefficients, we should use (see [16, chapter 1]) Schläfli's formula for the dual of a smooth plane cubic . The coordinates of a point satisfies to the sextic relation if and only if the line is tangent to the conic locus , where

Set . , and . .

*Proposition 3.9. *

*Proof. *Let , and set . Therefore,
Let
.

It is easy to prove that
The result follows from the summation of all .

*Proposition 3.10. *

*Proof. *Let , and set . Therefore,
Let
.

It is easy to prove that
The result follows from the summation of all , .

*Theorem 3.11. *

*Proof. *This result is a direct consequence of Propositions 3.9 and 3.10.

*Corollary 3.12. The Poincaré series of the algebras is
*

*Remark 3.13. *For , the case of Figure 1, our formula gives the same answer like the classical Pick's formula for integer points in a convex polygon with integer vertices on the plane ([17, chapter 10])
Here, and . In other hand the Pick's formula ingredients are
hence .

*This remark gives a good hint how one can use the developed machinery of integer points computations in rational polytopes to our problems.*

*4. -Invariant JPS in Any Dimension*

*4. -Invariant JPS in Any Dimension**In order to formulate the problem in any dimension, let us remember some number theoretic notions concerning the enumeration of nonnegative integer points in a polytope or more generally discrete volume of a polytope. *

*4.1. Enumeration of Integer Solutions to Linear Inequalities*

*4.1. Enumeration of Integer Solutions to Linear Inequalities**In their papers [18, 19], the authors study the problem of nonnegative integer solutions to linear inequalities as well as their relation with the enumeration of integer partitions and compositions.*

*Define the weight of a sequence of integers to be . If sequence of weight has all parts nonnegative, it is called a composition of ; if, in addition, is a nonincreasing sequence, we call it a partition of .*

*Given an integer matrix , , consider the set of nonnegative integer sequences satisfying the constraints
The associated full generating function is defined as follows:
This function “encapsulates” the solution set : the coefficient of in (, ) is a “listing” (as the terms of a polynomial) of all nonnegative integer solutions to (4.1) of weight , and the number of such solutions is the coefficient of in .*

*4.2. Formulation of the Problem in Any Dimension*

*4.2. Formulation of the Problem in Any Dimension**Let be the polynomial algebra with generators. For given polynomials , one can associate the JPS on given by
for .*

*We will denote by the particular Casimir of the Poisson structure . We suppose that each is homogeneous in the sense of -degree.*

*Proposition 4.1. Consider a JPS given by homogeneous (in the sense of -degree) polynomials . If is -invariant, then
where .*

*Proof. *Let , and consider the set , formed by the integers . We denote by the set of all permutation of elements of . We have
From the -degree condition,

We can deduce, therefore, that
And we obtain the first part of the result. The second part is the direct consequence of facts that
and the -degree condition.

*Set
Let be the set of all such that . One can easily check the following result.*

*Proposition 4.2. is a subvector space of . It is subalgebra of if .*

*We endow with the usual grading of the polynomial algebra . For , an element of , we denote by its usual weight degree. We denote by the homogeneous subspace of of degree .*

*Proposition 4.3. If is not a divisor of (in other words, ) then .*

*Proof. *It is clear the . We suppose now that . Let , . Then,
Hence,
Since , .

*Set . We suppose that . We want to find all such that and, therefore, the dimension as -vector space.*

*Proposition 4.4. There exist such that
*

*Proof. *That is, the direct consequence of the fact that .

*One can easily obtain the following result.*

*Proposition 4.5. The system equation (4.12) has as a solution
where and the satisfy the condition
*

*Therefore are completely determined by the set of nonnegative integer sequences satisfying the constraints
and such that
There are two approaches to determine the dimension of .*

*The first one is exactly as in the case of dimension 3. The constraint (4.16) is equivalent to say that *

Therefore, by replacing by this value, are completely determined by the set of nonnegative integer sequences satisfying the constraints Hence, the dimension is just the number of nonnegative integer points contained in the polytope given by the system (4.18), where .

In dimension 3, one obtains the triangle in given by the vertices , , and (see Section 3).

In dimension 4, we get the following polytope (see Figure 2).

For the second method, one can observe that the dimension of is nothing else that the cardinality of the set of all compositions of subjected to the constraints (4.15). Therefore, if is the set of all nonnegative integers satisfying the constraints (4.15) and is the associated generating function, then the dimension of is the coefficient of in .The set consists of all nonnegative integers points contained in the polytope of

(See Figure 3).

#### Acknowledgments

The authors are grateful to M. Beck and T. Schedler for useful and illuminative discussions. This work has begun when G. Ortenzi was visiting Mathematics Research Unit at Luxembourg. G. Ortenzi thanks this institute for the invitation and for the kind hospitality. A Part of this work has been done when S. R. T. Pelap: was visiting Max Planck Institute at Bonn. S. R. T. Pelap thanks this institute for the invitation and for good working conditions. S. R. T. Pelap have been partially financed by “Fond National de Recherche (Luxembourg)”. He is thankful to LAREMA for a kind invitation and a support during his stay in Angers. V. Rubtsov was partially supported by the French National Research Agency (ANR) Grant no. 2011 DIADEMS and by franco-ukrainian PICS (CNRS-NAS) in Mathematical Physics. He is grateful to MATPYL project for a support of T. Schedler visit in Angers and to the University of Luxembourg for a support of his visit to Luxembourg.

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