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Advances in Mathematical Physics
Volumeย 2011ย (2011), Article IDย 252186, 18 pages
http://dx.doi.org/10.1155/2011/252186
Research Article

Integer Solutions of Integral Inequalities and ๐ป-Invariant Jacobian Poisson Structures

1Dipartimento di Matematica Pura e Applicazioni, Universitร  degli Milano Bicocca, Via R.Cozzi 53, 20125 Milano, Italy
2Dรฉpartement de Mathรฉmatiques 2, Laboratoire Angevin de Recherche en Mathรฉmatiques Universitรฉ D'Angers, boulevard Lavoisier, 49045 Angers, France
3Mathematics 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-Zhongย Zhang

Copyright ยฉ 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 ๐‘› variables โ„‚[๐‘ฅ0,๐‘ฅ1,โ€ฆ,๐‘ฅ๐‘›โˆ’1] over โ„‚ and the action of some subgroup ๐ป๐‘› of ๐บ๐ฟ๐‘›(โ„‚) generated by the shifts operators ๐‘ฅ๐‘–โ†’๐‘ฅ๐‘–+1(modโ„ค๐‘›) and by the operators ๐‘ฅโ†’๐œ€๐‘–๐‘ฅ๐‘–, where ๐œ€๐‘›=1. We are interested in the polynomial Poisson brackets on โ„‚[๐‘ฅ0,๐‘ฅ1,โ€ฆ,๐‘ฅ๐‘›โˆ’1] which are โ€œstableโ€ 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 โ€œquasiclassical limitsโ€ 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 ๐‘› variables. The above-mentioned Heisenberg group action provides the automorphisms of Sklyanin algebras which are compatible with the grading and defines an ๐ป๐‘›-action on the elliptic quadratic Poisson structures on โ„™๐‘›. The latter are identified with Poisson structures on some moduli spaces of the degree ๐‘› and rank ๐‘˜+1 vector bundles with parabolic structure (= the flag 0โŠ‚๐นโŠ‚โ„‚๐‘˜+1 on the elliptic curve โ„ฐ). We will denote this elliptic Poisson algebras by ๐‘ž๐‘›;๐‘˜(โ„ฐ). The algebras ๐‘ž๐‘›;๐‘˜(๐ธ) arise in the Feigin-Odesskii โ€œdeformationalโ€ 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 {๐‘ƒ,๐‘„} of two polynomials ๐‘ƒ and ๐‘„ is given by the determinant of Jacobi matrix of functions (๐‘ƒ,๐‘„,๐‘ƒ1,โ€ฆ,๐‘ƒ๐‘›โˆ’2). The polynomials ๐‘ƒ๐‘–, 1โ‰ค๐‘–โ‰ค๐‘›โˆ’2 are Casimirs of the bracket and under some mild condition of independence are generators of the centrum for the Jacobian Poisson algebra structure on โ„‚[๐‘ฅ0,โ€ฆ,๐‘ฅ๐‘›โˆ’1]. 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, 5โ€“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 (๐‘›=3) and Sklyanin (๐‘›=4) algebras which are both elliptic and Jacobian. It is no longer true for ๐‘›>4. The relations between the Sklyanin Poisson algebras ๐‘ž๐‘›,๐‘˜(โ„ฐ) whose centrum has dimension 1 (for ๐‘› odd) and 2 (for ๐‘› even) in the case ๐‘˜=1 and is generated by ๐‘™=๐”ค๐‘๐‘‘(๐‘›,๐‘˜+1) Casimirs for ๐‘ž๐‘›,๐‘˜(โ„ฐ) for ๐‘˜>1 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 ๐‘›=5 was remarked in [8] and was attributed to so-called Briesckorn-Pham polynomials for ๐‘›=5๐‘ƒ1=4๎“๐‘–=0๐›ผ๐‘–๐‘ฅ๐‘–;๐‘ƒ2=4๎“๐‘–=0๐›ฝ๐‘–๐‘ฅ2๐‘–;๐‘ƒ3=4๎“๐‘–=0๐›พ๐‘–๐‘ฅ2๐‘–.(1.1) It is easy to check that the homogeneous quintic ๐‘ƒ=๐‘ƒ1๐‘ƒ2๐‘ƒ3 (see Section 4.2) defines a Casimir for some rational degeneration of (one of) elliptic algebras ๐‘ž5,1(โ„ฐ) and ๐‘ž5,2(โ„ฐ) if it satisfies the ๐ป-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 โ„‹ of polynomial algebra. This vector space is completely determined by some enumerative problem of a number-theoretic type. More precisely, the homogeneous subspace โ„‹๐‘– of โ„‹ of degree ๐‘– is in bijection with integer solutions of a system of Diophant inequalities. Geometric interpretation of the dimension of โ„‹๐‘– is described in terms of integer points in a convex polytope given by this Diophant system. In the special case of dimension 3, โ„‹ 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 ๐ป-invariant not necessarily quadratic Jacobian Poisson algebras with three generators. As a byproduct, we explicitly compute the Poincarรฉ series of โ„‹. In this dimension, we observe that the ๐ป-invariant JPS of degree 5 is given by the Casimir sextic ๐‘ƒโˆจ=16๐‘Ž๎€ท๐‘ฅ60+๐‘ฅ61+๐‘ฅ62๎€ธ+13๐‘๎€ท๐‘ฅ30๐‘ฅ31+๐‘ฅ30๐‘ฅ32+๐‘ฅ31๐‘ฅ32๎€ธ๎€ท๐‘ฅ+๐‘40๐‘ฅ1๐‘ฅ2+๐‘ฅ0๐‘ฅ41๐‘ฅ2+๐‘ฅ0๐‘ฅ1๐‘ฅ42๎€ธ+12๐‘‘๐‘ฅ20๐‘ฅ21๐‘ฅ22,(1.2)๐‘Ž,๐‘,๐‘,๐‘‘โˆˆโ„‚. This structure is a โ€œprojectively dualโ€ to the Artin-Schelter-Tate elliptic Poisson structure which is the ๐ป-invariant JPS given by the cubic ๎€ท๐‘ฅ๐‘ƒ=30+๐‘ฅ31+๐‘ฅ32๎€ธ+๐›พ๐‘ฅ0๐‘ฅ1๐‘ฅ2,(1.3) where ๐›พโˆˆโ„‚. In fact, the algebraic variety โ„ฐโˆจโˆถ{๐‘ƒโˆจ=0}โˆˆโ„™2 is the (generically) projectively dual to the elliptic curve โ„ฐโˆถ{๐‘ƒ=0}โŠ‚โ„™2.

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, ๐พ 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 โ„› be a commutative ๐พ-algebra. One says that โ„› is a Poisson algebra if โ„› is endowed with a Lie bracket, indicated with {โ‹…,โ‹…}, which is also a biderivation. One can also say that โ„› is endowed with a Poisson structure, and therefore, the bracket {โ‹…,โ‹…} is called the Poisson bracket. Elements of the center are called Casimirs: ๐‘Žโˆˆโ„› is a Casimir if {๐‘Ž,๐‘}=0 for all ๐‘โˆˆโ„›.

A Poisson manifold ๐‘€ (smooth, algebraic, etc.) is a manifold whose function algebra ๐’œ (๐ถโˆž(๐‘€), 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 ๐‘ž1=12๎€ท๐‘ฅ20+๐‘ฅ22๎€ธ+๐‘˜๐‘ฅ1๐‘ฅ3,๐‘ž2=12๎€ท๐‘ฅ21+๐‘ฅ23๎€ธ+๐‘˜๐‘ฅ0๐‘ฅ2(2.1) be two elements of โ„‚[๐‘ฅ0,๐‘ฅ1,๐‘ฅ2,๐‘ฅ3], where ๐‘˜โˆˆโ„‚.

On โ„‚[๐‘ฅ0,๐‘ฅ1,๐‘ฅ2,๐‘ฅ3], a Poisson structure ๐œ‹ is defined by {๐‘“,๐‘”}๐œ‹โˆถ=๐‘‘๐‘“โˆง๐‘‘๐‘”โˆง๐‘‘๐‘ž1โˆง๐‘‘๐‘ž2๐‘‘๐‘ฅ0โˆง๐‘‘๐‘ฅ1โˆง๐‘‘๐‘ฅ2โˆง๐‘‘๐‘ฅ3,(2.2) or more explicitly (mod โ„ค4) ๎€ฝ๐‘ฅ๐‘–,๐‘ฅ๐‘–+1๎€พ=๐‘˜2๐‘ฅ๐‘–๐‘ฅ๐‘–+1โˆ’๐‘ฅ๐‘–+2๐‘ฅ๐‘–+3,๎€ฝ๐‘ฅ๐‘–,๐‘ฅ๐‘–+2๎€พ๎€ท๐‘ฅ=๐‘˜2๐‘–+3โˆ’๐‘ฅ2๐‘–+1๎€ธ,๐‘–=0,1,2,3.(2.3) 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 โ€œCasimirsโ€. 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 โ„ฐ given by two quadrics ๐‘ž1,2. We can think about this curve โ„ฐ as a complete intersection of the couple ๐‘ž1=0, ๐‘ž2=0 embedded in ๐’ž๐‘ƒ3 (as it was observed in Sklyanin's initial paper).

A possible generalization one can be obtained considering ๐‘›โˆ’2 polynomials ๐‘„๐‘– in ๐พ๐‘› with coordinates ๐‘ฅ๐‘–, ๐‘–=0,โ€ฆ,๐‘›โˆ’1. We can define a bilinear differential operation ๎€บ๐‘ฅ{โ‹…,โ‹…}โˆถ๐พ1,โ€ฆ,๐‘ฅ๐‘›๎€ป๎€บ๐‘ฅโŠ—๐พ1,โ€ฆ,๐‘ฅ๐‘›๎€ป๎€บ๐‘ฅโŸถ๐พ1,โ€ฆ,๐‘ฅ๐‘›๎€ป,(2.4) by{๐‘“,๐‘”}=๐‘‘๐‘“โˆง๐‘‘๐‘”โˆง๐‘‘๐‘„1โˆงโ‹ฏโˆง๐‘‘๐‘„๐‘›โˆ’2๐‘‘๐‘ฅ1โˆง๐‘‘๐‘ฅ2โˆงโ‹ฏโˆง๐‘‘๐‘ฅ๐‘›๎€บ๐‘ฅ,๐‘“,๐‘”โˆˆ๐พ1,โ€ฆ,๐‘ฅ๐‘›๎€ป.(2.5) This operation, which gives a Poisson algebra structure on ๐พ[๐‘ฅ1,โ€ฆ,๐‘ฅ๐‘›], is called a Jacobian Poisson structure (JPS), and it is a partial case of more general ๐‘›โˆ’๐‘š-ary Nambu operation given by an antisymmetric ๐‘›โˆ’๐‘š-polyvector field introduced by Nambu [6] and was extensively studied by Takhtajan [5].

The polynomials ๐‘„๐‘–,โ€‰โ€‰๐‘–=1,โ€ฆ,๐‘›โˆ’2 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 ๐‘ž๐‘›(โ„ฐ,๐œ‚) and ๐‘ž๐‘›,๐‘˜(โ„ฐ,๐œ‚)

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

These algebras, defined by Feฤญgin and Odesskiฤญ, arise as quasiclassical limits of elliptic associative algebras ๐‘„๐‘›(โ„ฐ,๐œ‚) and ๐‘„๐‘›,๐‘˜(โ„ฐ,๐œ‚) [12, 13].

Let ฮ“=โ„ค+๐œโ„คโŠ‚โ„‚ be an integral lattice generated by 1 and ๐œโˆˆโ„‚, with Im๐œ>0. Consider the elliptic curve โ„ฐ=โ„‚/ฮ“ and a point ๐œ‚ on this curve.

In their article [13], given ๐‘˜<๐‘›, mutually prime, Odesskiฤญ and Feฤญgin construct an algebra, called elliptic, ๐‘„๐‘›,๐‘˜(โ„ฐ,๐œ‚), as an algebra defined by ๐‘› generators {๐‘ฅ๐‘–,๐‘–โˆˆโ„ค/๐‘›โ„ค} and the following relations:๎“๐‘Ÿโˆˆโ„ค/๐‘›โ„ค๐œƒ๐‘—โˆ’๐‘–+๐‘Ÿ(๐‘˜โˆ’1)(0)๐œƒ๐‘˜๐‘Ÿ(๐œ‚)๐œƒ๐‘—โˆ’๐‘–โˆ’๐‘Ÿ๐‘ฅ(โˆ’๐œ‚)๐‘—โˆ’๐‘Ÿ๐‘ฅ๐‘–+๐‘Ÿโ„ค=0,๐‘–โ‰ ๐‘—,๐‘–,๐‘—โˆˆ,๐‘›โ„ค(2.6) where ๐œƒ๐›ผ are theta functions [13].

These family of algebras has the following properties: (1)the center of the algebra ๐‘„๐‘›,๐‘˜(โ„ฐ,๐œ‚), for generic โ„ฐ and ๐œ‚, is the algebra of polynomial of ๐‘š=๐‘๐‘”๐‘๐‘‘(๐‘›,๐‘˜+1) variables of degree ๐‘›/๐‘š,(2)๐‘„๐‘›,๐‘˜(โ„ฐ,0)=โ„‚[๐‘ฅ1,โ€ฆ,๐‘ฅ๐‘›] is commutative, (3)๐‘„๐‘›,๐‘›โˆ’1(โ„ฐ,๐œ‚)=โ„‚[๐‘ฅ1,โ€ฆ,๐‘ฅ๐‘›] is commutative for all ๐œ‚, (4)๐‘„๐‘›,๐‘˜(โ„ฐ,๐œ‚)โ‰ƒ๐‘„๐‘›,๐‘˜โ€ฒ(โ„ฐ,๐œ‚), if ๐‘˜๐‘˜๎…žโ‰ก1 (mod ๐‘›), (5)the maps ๐‘ฅ๐‘–โ†ฆ๐‘ฅ๐‘–+1 et ๐‘ฅ๐‘–โ†ฆ๐œ€๐‘–๐‘ฅ๐‘–, where ๐œ€๐‘›=1, define automorphisms of the algebra ๐‘„๐‘›,๐‘˜(โ„ฐ,๐œ‚),(6)the algebras ๐‘„๐‘›,๐‘˜(โ„ฐ,๐œ‚) are deformations of polynomial algebras. The associated Poisson structure is denoted by ๐‘ž๐‘›,๐‘˜(โ„ฐ,๐œ‚), (7)among the algebras ๐‘ž๐‘›,๐‘˜(โ„ฐ,๐œ‚), only ๐‘ž3(โ„ฐ,๐œ‚) (the Artin-Schelter-Tate algebra) and the Sklyanin algebra ๐‘ž4(โ„ฐ,๐œ‚) are Jacobian Poisson structures.

2.3. The Heisenberg Invariant Poisson Structures
2.3.1. The G-Invariant Poisson Structures

Let ๐บ be a group acting on a Poisson algebra โ„›.

Definition 2.1. A Poisson bracket {โ‹…,โ‹…} on โ„› is said to be a ๐บ-invariant if ๐บ acts on โ„› by Poisson automorphisms.
In other words, for every ๐‘”โˆˆ๐บ, the morphism ๐œ‘๐‘”โˆถโ„›โ†’โ„›, ๐‘Žโ†ฆ๐‘”โ‹…๐‘Ž is an automorphism and the following diagram is a commutative:

252186.equation.3(2.7)

2.3.2. The H-Invariant Poisson Structures

In their paper [2], the authors introduced the notion of ๐ป-invariant Poisson structures. That is, a special case of a ๐บ-invariant structure when ๐บ in the finite Heisenberg group and โ„› is the polynomial algebra. Let us remember this notion.

Let ๐‘‰ be a complex vector space of dimension ๐‘› and ๐‘’0,โ€ฆ,๐‘’๐‘›โˆ’1 a basis of ๐‘‰. Take the ๐‘›th primitive root of unity ๐œ€=๐‘’2๐œ‹๐‘–/๐‘›.

Consider ๐œŽ,๐œ of ๐บ๐ฟ(๐‘‰) defined by ๐œŽ๎€ท๐‘’๐‘š๎€ธ=๐‘’๐‘šโˆ’1,๐œ๎€ท๐‘’๐‘š๎€ธ=๐œ€๐‘š๐‘’๐‘š.(2.8) The Heisenberg of dimension ๐‘› is nothing else that the subspace ๐ป๐‘›โŠ‚๐บ๐ฟ(๐‘‰) generated by ๐œŽ and ๐œ.

From now on, we assume that ๐‘‰=โ„‚๐‘›, with ๐‘ฅ0,๐‘ฅ1,โ€ฆ,๐‘ฅ๐‘›โˆ’1 as a basis and consider the coordinate ring โ„‚[๐‘ฅ0,๐‘ฅ1,โ€ฆ,๐‘ฅ๐‘›โˆ’1].

Naturally, ๐œŽ and ๐œ act by automorphisms on the algebra โ„‚[๐‘ฅ0,๐‘ฅ1,โ€ฆ,๐‘ฅ๐‘›โˆ’1] as follows: ๎€ท๐œŽโ‹…๐›ผ๐‘ฅ๐›ผ00๐‘ฅ๐›ผ11โ‹ฏ๐‘ฅ๐›ผ๐‘›โˆ’1๐‘›โˆ’1๎€ธ=๐›ผ๐‘ฅ๐›ผ๐‘›โˆ’10๐‘ฅ๐›ผ01โ‹ฏ๐‘ฅ๐›ผ๐‘›โˆ’2๐‘›โˆ’1,๎€ท๐œโ‹…๐›ผ๐‘ฅ๐›ผ00๐‘ฅ๐›ผ11โ‹ฏ๐‘ฅ๐›ผ๐‘›โˆ’1๐‘›โˆ’1๎€ธ=๐œ€๐›ผ1+2๐›ผ2+โ‹ฏ+(๐‘›โˆ’1)๐›ผ๐‘›โˆ’1๐›ผ๐‘ฅ๐›ผ00๐‘ฅ๐›ผ11โ‹ฏ๐‘ฅ๐›ผ๐‘›โˆ’1๐‘›โˆ’1.(2.9) We introduced in [2] the notion of ๐œ-degree on the polynomial algebra โ„‚[๐‘ฅ0,๐‘ฅ1,โ€ฆ,๐‘ฅ๐‘›โˆ’1]. The ๐œ-degree of a monomial ๐‘€=๐›ผ๐‘ฅ๐›ผ00๐‘ฅ๐›ผ11โ‹ฏ๐‘ฅ๐›ผ๐‘›โˆ’1๐‘›โˆ’1 is the positive integer ๐›ผ1+2๐›ผ2+โ‹ฏ+(๐‘›โˆ’1)๐›ผ๐‘›โˆ’1โˆˆโ„ค/๐‘›โ„ค if ๐›ผโ‰ 0 and โˆ’โˆž if not. The ๐œ-degree of ๐‘€ is denoted ๐œ๐œ›(๐‘€). A ๐œ-degree of a polynomial is the highest ๐œ-degree of its monomials.

For simplicity, the ๐ป๐‘›-invariance condition will be referred from now on just as ๐ป-invariance. An ๐ป-invariant Poisson bracket on ๐’œ=โ„‚[๐‘ฅ0,๐‘ฅ1,โ€ฆ,๐‘ฅ๐‘›โˆ’1] is nothing but a bracket on ๐’œ which satisfy the following: ๎€ฝ๐‘ฅ๐‘–+1,๐‘ฅ๐‘—+1๎€พ๎€ฝ๐‘ฅ=๐œŽโ‹…๐‘–,๐‘ฅ๐‘—๎€พ,๎€ฝ๐‘ฅ๐œโ‹…๐‘–,๐‘ฅ๐‘—๎€พ=๐œ€๐‘–+๐‘—๎€ฝ๐‘ฅ๐‘–,๐‘ฅ๐‘—๎€พ,(2.10) for all ๐‘–,๐‘—โˆˆโ„ค/๐‘›โ„ค.

The ๐œ invariance is, in some sense, a โ€œdiscreteโ€ homogeneity.

Proposition 2.2 (see [2]). The Sklyanin-Odesskii-Feigin Poisson algebras ๐‘ž๐‘›,๐‘˜(โ„ฐ) are ๐ป-invariant Poisson algebras.

Therefore, an ๐ป-invariant Poisson structures on the polynomial algebra โ„› 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 ๐ป-invariant Poisson structures.

Proposition 2.3 (see [2]). If {โ‹…,โ‹…} is an ๐ป-invariant polynomial Poisson bracket, the usual polynomial degree of the monomial of {๐‘ฅ๐‘–,๐‘ฅ๐‘—} equals to 2+๐‘ ๐‘›,โ€‰โ€‰๐‘ โˆˆโ„•.

Proposition 2.4 (see [2]). Let ๐‘ƒโˆˆโ„›=โ„‚[๐‘ฅ0,โ€ฆ,๐‘ฅ๐‘›โˆ’1].
For all ๐‘–โˆˆ{0,โ€ฆ,๐‘›โˆ’1}, ๐œŽโ‹…๐œ•๐น๐œ•๐‘ฅ๐‘–=๐œ•(๐œŽโ‹…๐น)๐œ•๎€ท๐œŽโ‹…๐‘ฅ๐‘–๎€ธ.(2.11)

3. ๐ป-Invariant JPS in Dimension 3

We consider first a generalization of Artin-Schelter-Tate quadratic Poisson algebras. Let โ„›=โ„‚[๐‘ฅ0,๐‘ฅ1,๐‘ฅ2] be the polynomial algebra with 3 generators. For every ๐‘ƒโˆˆ๐’œ, we have a JPS ๐œ‹(๐‘ƒ) on โ„› given by ๎€ฝ๐‘ฅ๐‘–,๐‘ฅ๐‘—๎€พ=๐œ•๐‘ƒ๐œ•๐‘ฅ๐‘˜,(3.1) where (๐‘–,๐‘—,๐‘˜)โˆˆโ„ค/3โ„ค is a cyclic permutation of (0,1,2). Let โ„‹ be the set of all ๐‘ƒโˆˆ๐’œ such that ๐œ‹(๐‘ƒ) is an ๐ป-invariant Poisson structure.

Proposition 3.1. If ๐‘ƒโˆˆโ„‹ is a homogeneous polynomial, then ๐œŽโ‹…๐‘ƒ=๐‘ƒ and ๐œ๐œ›(๐‘ƒ)=0.

Proof. Let (๐‘–,๐‘—,๐‘˜)โˆˆ(โ„ค/3โ„ค)3 be a cyclic permutation of (0,1,2). One has ๎€ฝ๐‘ฅ๐‘–,๐‘ฅ๐‘—๎€พ=๐œ•๐‘ƒ๐œ•๐‘ฅ๐‘˜,(3.2)๎€ฝ๐‘ฅ๐‘–+1,๐‘ฅ๐‘—+1๎€พ=๐œ•๐‘ƒ๐œ•๐‘ฅ๐‘˜+1.(3.3) Using Proposition 2.4, we conclude that for all ๐‘šโˆˆโ„ค/3โ„ค,โ€‰โ€‰๐œ•๐œŽ๐‘ƒ/๐œ•๐‘ฅ๐‘š=๐œ•๐‘ƒ/๐œ•๐‘ฅ๐‘š.
It gives that ๐œŽโ‹…๐‘ƒ=๐‘ƒ.
On the other hand, from (3.2), one has ๐œโˆ’๐œ›(๐‘ƒ)โ‰ก๐‘–+๐‘—+๐‘˜mod3. And we get the second half of the proposition.

Proposition 3.2. โ„‹ is a subalgebra of โ„›.

Proof. Let ๐‘ƒ,๐‘„โˆˆโ„‹. It is clear that for all ๐›ผ,๐›ฝโˆˆโ„‚,โ€‰โ€‰๐›ผ๐‘ƒ+๐›ฝ๐‘„ belongs to โ„‹.
Let us denote by {โ‹…,โ‹…}๐น the JPS associated with the polynomial ๐นโˆˆโ„›. It is easy to verify that {๐‘ฅ๐‘–,๐‘ฅ๐‘—}๐‘ƒ๐‘„=๐‘ƒ{๐‘ฅ๐‘–,๐‘ฅ๐‘—}๐‘„+๐‘„{๐‘ฅ๐‘–,๐‘ฅ๐‘—}๐‘ƒ. Therefore, it is clear that the ๐ป-invariance condition is verified for the JPS associated to the polynomial ๐‘ƒ๐‘„.

We endow โ„‹ with the usual grading of the polynomial algebra โ„›. For ๐น, an element of โ„›, we denote by ๐œ›(๐น) its usual weight degree. We denote by โ„‹๐‘– the homogeneous subspace of โ„‹ of degree ๐‘–.

Proposition 3.3. If 3 does not divide ๐‘–โˆˆโ„• (in other words ๐‘–โ‰ 3๐‘˜), then โ„‹๐‘–=0.

Proof. First of all โ„‹0=โ„‚. We suppose now that ๐‘–โ‰ 0. Let ๐‘ƒโˆˆโ„‹๐‘–,๐‘ƒโ‰ 0. Then, ๐œ›(๐‘ƒ)=๐‘–. It follows from Proposition 2.3 and the definition of the Poisson brackets that there exists ๐‘ โˆˆโ„• such that ๐œ›(๐‘ฅ๐‘–,๐‘ฅ๐‘—)=2+3๐‘ . The result follows from the fact that ๐œ›(๐‘ฅ๐‘–,๐‘ฅ๐‘—)=๐œ›(๐‘ƒ)โˆ’1.

Set โˆ‘๐‘๐‘ƒ=๐›ผ๐‘ฅ๐›ผ00๐‘ฅ๐›ผ11๐‘ฅ๐›ผ22, where ๐›ผ=(๐›ผ0,๐›ผ1,๐›ผ2). We suppose that ๐œ›(๐‘ƒ)=3(1+๐‘ ). We want to find all ๐›ผ0,๐›ผ1,๐›ผ2 such that ๐‘ƒโˆˆโ„‹ and, therefore, the dimension โ„‹3(1+๐‘ ) as โ„‚-vector space.

Proposition 3.4. There exist ๐‘ ๎…ž, ๐‘ ๎…ž๎…ž, and ๐‘ ๎…ž๎…ž๎…ž such that ๐›ผ0+๐›ผ1+๐›ผ2=3(1+๐‘ ),0๐›ผ0+๐›ผ1+2๐›ผ2=3๐‘ ๎…ž,๐›ผ0+2๐›ผ1+0๐›ผ2=3๐‘ ๎…ž๎…ž,2๐›ผ0+0๐›ผ1+1๐›ผ2=3๐‘ ๎…ž๎…ž๎…ž.(3.4)

Proof. This is a direct consequence of Proposition 3.3.

Proposition 3.5. The system equation (3.4) has as solutions the following set: ๐›ผ0=4๐‘Ÿโˆ’2๐‘ ๎…žโˆ’๐‘ ๎…ž๎…ž,๐›ผ1=โˆ’2๐‘Ÿ+๐‘ ๎…ž+2๐‘ ๎…ž๎…ž,๐›ผ2=๐‘Ÿ+๐‘ ๎…žโˆ’๐‘ ๎…ž๎…ž,(3.5) where ๐‘Ÿ=1+๐‘ ,โ€‰โ€‰๐‘ ๎…ž and ๐‘ ๎…ž๎…ž live in the polygon given by the following inequalities in โ„2โˆถ๐‘ฅ+๐‘ฆโฉฝ3๐‘Ÿ,2๐‘ฅ+๐‘ฆโฉฝ4๐‘Ÿ,โˆ’๐‘ฅโˆ’2๐‘ฆโฉฝโˆ’2๐‘Ÿ,โˆ’๐‘ฅ+๐‘ฆโฉฝ๐‘Ÿ.(3.6)

Remark 3.6. For ๐‘Ÿ=1, one obtains the Artin-Schelter-Tate Poisson algebra which is the JPS given by the Casimir ๐‘ƒ=๐›ผ(๐‘ฅ30+๐‘ฅ31+๐‘ฅ32)+๐›ฝ๐‘ฅ0๐‘ฅ1๐‘ฅ2,โ€‰โ€‰๐›ผ,๐›ฝโˆˆโ„‚. Suppose that ๐›ผโ‰ 0, then it can take the form ๎€ท๐‘ฅ๐‘ƒ=30+๐‘ฅ31+๐‘ฅ32๎€ธ+๐›พ๐‘ฅ0๐‘ฅ1๐‘ฅ2,(3.7) where ๐›พโˆˆโ„‚. The interesting feature of this Poisson algebra is that their polynomial character is preserved even after the following nonalgebraic changes of variables. Let ๐‘ฆ0=๐‘ฅ0;๐‘ฆ1=๐‘ฅ1๐‘ฅ2โˆ’1/2;๐‘ฆ2=๐‘ฅ23/2.(3.8) The polynomial ๐‘ƒ in the coordinates (๐‘ฆ0;๐‘ฆ1;๐‘ฆ2) has the form ๎‚๎€ท๐‘ฆ๐‘ƒ=30+๐‘ฆ31๐‘ฆ2+๐‘ฆ22๎€ธ+๐›พ๐‘ฆ0๐‘ฆ1๐‘ฆ2.(3.9) The Poisson bracket is also polynomial (which is not evident at all!) and has the same form ๎€ฝ๐‘ฅ๐‘–,๐‘ฅ๐‘—๎€พ=๐œ•๎‚๐‘ƒ๐œ•๐‘ฅ๐‘˜,(3.10) where (๐‘–,๐‘—,๐‘˜) is the cyclic permutation of (0,1,2). This JPS structure is no longer satisfying the Heisenberg invariance condition. But it is invariant with respect the following toric action: (โ„‚โˆ—)3ร—โ„™2โ†’โ„™ given by ๎€ท๐‘ฅ๐œ†โ‹…0โˆถ๐‘ฅ1โˆถ๐‘ฅ2๎€ธ=๎€ท๐œ†2๐‘ฅ0โˆถ๐œ†๐‘ฅ1โˆถ๐œ†3๐‘ฅ2๎€ธ.(3.11)
Put deg๐‘ฆ0=2;deg๐‘ฆ2=1;deg๐‘ฆ2=3. Then, the polynomial ๎‚๐‘ƒ is also homogeneous in (๐‘ฆ0;๐‘ฆ1;๐‘ฆ2) and defines an elliptic curve ๎‚๐‘ƒ=0 in the weighted projective space ๐•Ž๐‘ƒ2;1;3.
The similar change of variables ๐‘ง0=๐‘ฅ0โˆ’3/4๐‘ฅ13/2;๐‘ง1=๐‘ฅ01/4๐‘ฅ1โˆ’1/2๐‘ฅ2;๐‘ง2=๐‘ฅ03/2(3.12) defines the JPS structure invariant with respect to the torus action (โ„‚โˆ—)3ร—โ„™2โ†’โ„™ given by ๎€ท๐‘ฅ๐œ†โ‹…0โˆถ๐‘ฅ1โˆถ๐‘ฅ2๎€ธ=๎€ท๐œ†๐‘ฅ0โˆถ๐œ†๐‘ฅ1โˆถ๐œ†2๐‘ฅ2๎€ธ,(3.13) and related to the elliptic curve 1/3(๐‘ง22+๐‘ง20๐‘ง2+๐‘ง0๐‘ง31)+๐‘˜๐‘ง0๐‘ง1๐‘ง2=0 in the weighted projective space ๐•Ž๐‘ƒ1;1;2.
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 โ„2 given by the system (3.6) is a triangle ๐’ฏ๐‘Ÿ with (0,๐‘Ÿ),โ€‰โ€‰(๐‘Ÿ,2๐‘Ÿ), and (2๐‘Ÿ,0) as vertices. Then, dimโ„‹3๐‘Ÿ=Card(๐’ฏ๐‘Ÿโˆฉโ„•2).

Remark 3.8. For ๐‘Ÿ=2, the case of Figure 1, the generic Heisenberg-invariant JPS is given by the sextic polynomial ๐‘ƒโˆจ=16๐‘Ž๎€ท๐‘60+๐‘61+๐‘62๎€ธ+13๐‘๎€ท๐‘30๐‘31+๐‘30๐‘32+๐‘31๐‘32๎€ธ๎€ท๐‘+๐‘40๐‘1๐‘2+๐‘0๐‘41๐‘2+๐‘0๐‘1๐‘42๎€ธ+12๐‘‘๐‘20๐‘21๐‘22,๐‘Ž,๐‘,๐‘,๐‘‘โˆˆโ„‚.(3.14) The corresponding Poisson bracket takes the form ๎€ฝ๐‘ฅ๐‘–,๐‘ฅ๐‘—๎€พ=๐‘Ž๐‘ฅ5๐‘˜๎‚€๐‘ฅ+๐‘3๐‘–+๐‘ฅ3๐‘—๎‚๐‘ฅ2๐‘˜+๐‘๐‘ฅ๐‘–๐‘ฅ๐‘—๎‚€๐‘ฅ3๐‘–+๐‘ฅ3๐‘—+4๐‘ฅ3๐‘˜๎‚+๐‘‘๐‘ฅ2๐‘–๐‘ฅ2๐‘—๐‘ฅ๐‘˜,(3.15) where ๐‘–,๐‘—,๐‘˜ are the cyclic permutations of 0,1,2.
This new JPS should be considered as the โ€œprojectively dualโ€ to the Artin-Schelter-Tate JPS, since the algebraic variety โ„ฐโˆจโˆถ๐‘ƒโˆจ=0 is generically the projective dual curve in โ„™2 to the elliptic curve ๎€ท๐‘ฅโ„ฐโˆถ๐‘ƒ=30+๐‘ฅ31+๐‘ฅ32๎€ธ+๐›พ๐‘ฅ0๐‘ฅ1๐‘ฅ2=0.(3.16)
To establish the exact duality and the explicit values of the coefficients, we should use (see [16, chapter 1]) Schlรคfli's formula for the dual of a smooth plane cubic โ„ฐ=0โŠ‚โ„™2. The coordinates (๐‘0โˆถ๐‘1โˆถ๐‘2)โˆˆโ„™2โˆ— of a point ๐‘โˆˆโ„™2โˆ— satisfies to the sextic relation โ„ฐโˆจ=0 if and only if the line ๐‘ฅ0๐‘0+๐‘ฅ1๐‘1+๐‘ฅ2๐‘2=0 is tangent to the conic locus ๐’ž(๐‘ฅ,๐‘)=0, where ๐’ž||||||||||||||||(๐‘ฅ,๐‘)=0๐‘0๐‘1๐‘2๐‘0๐œ•2๐‘ƒ๐œ•๐‘ฅ20๐œ•2๐‘ƒ๐œ•๐‘ฅ0๐œ•๐‘ฅ1๐œ•2๐‘ƒ๐œ•๐‘ฅ0๐œ•๐‘ฅ2๐‘1๐œ•2๐‘ƒ๐œ•๐‘ฅ1๐œ•๐‘ฅ0๐œ•2๐‘ƒ๐œ•๐‘ฅ1๐œ•๐‘ฅ1๐œ•2๐‘ƒ๐œ•๐‘ฅ1๐œ•๐‘ฅ2๐‘2๐œ•2๐‘ƒ๐œ•๐‘ฅ2๐œ•๐‘ฅ0๐œ•2๐‘ƒ๐œ•๐‘ฅ2๐œ•๐‘ฅ1๐œ•2๐‘ƒ๐œ•๐‘ฅ2๐œ•๐‘ฅ2||||||||||||||||.(3.17)
Set ๐’ฎ๐‘Ÿ=๐’ฏ๐‘Ÿโˆฉโ„•2. โ€‰๐’ฎ๐‘Ÿ=๐’ฎ1๐‘Ÿโˆช๐’ฎ2๐‘Ÿ, ๐’ฎ1๐‘Ÿ={(๐‘ฅ,๐‘ฆ)โˆˆ๐’ฎ๐‘Ÿโˆถ0โ‰ค๐‘ฅโ‰ค๐‘Ÿ} and ๐’ฎ1๐‘Ÿ={(๐‘ฅ,๐‘ฆ)โˆˆ๐’ฎ๐‘Ÿโˆถ๐‘Ÿ<๐‘ฅโ‰ค2๐‘Ÿ}. โ€‰dimโ„‹3๐‘Ÿ=Card(๐’ฎ1๐‘Ÿ)+Card(๐’ฎ2๐‘Ÿ).

252186.fig.001
Figure 1: An example of the triangle ๐’ฏ2 in the case ๐‘Ÿ=2.

Proposition 3.9. Card๎€ท๐’ฎ1๐‘Ÿ๎€ธ=โŽงโŽชโŽจโŽชโŽฉ3๐‘Ÿ2+6๐‘Ÿ+44if๐‘Ÿiseven,3๐‘Ÿ2+6๐‘Ÿ+34if๐‘Ÿisodd.(3.18)

Proof. Let ๐›ผโˆˆ{0,โ€ฆ,๐‘Ÿ}, and set ๐’Ÿ๐›ผ๐‘Ÿ={(๐‘ฅ,๐‘ฆ)โˆˆ๐’ฎ1๐‘Ÿโˆถ๐‘ฅ=๐›ผ}. Therefore, Card๎€ท๐’ฎ1๐‘Ÿ๎€ธ=๐‘Ÿ๎“๐›ผ=0Card๎€ท๐’Ÿ๐›ผ๐‘Ÿ๎€ธ.(3.19) Let ๐›ฝ๐›ผmax๎€ฝ=max๐›ฝโˆถ(๐›ผ,๐›ฝ)โˆˆ๐’Ÿ๐›ผ๐‘Ÿ๎€พ,๐›ฝ๐›ผmin๎€ฝ=min๐›ฝโˆถ(๐›ผ,๐›ฝ)โˆˆ๐’Ÿ๐›ผ๐‘Ÿ๎€พ.(3.20)Card(๐’Ÿ๐›ผ๐‘Ÿ)=๐›ฝ๐›ผmaxโˆ’๐›ฝ๐›ผmin+1.
It is easy to prove that Card๎€ท๐’Ÿ๐›ผ๐‘Ÿ๎€ธ๎‚ž๐›ผ=(๐›ผ+1)+2๎‚Ÿ=โŽงโŽชโŽจโŽชโŽฉ3๐›ผ+22if๐›ผiseven,3๐›ผ+12if๐›ผisodd.(3.21) The result follows from the summation of all Card(๐’Ÿ๐›ผ๐‘Ÿ),๐›ผโˆˆ{0,โ€ฆ,๐‘Ÿ}.

Proposition 3.10. Card๎€ท๐’ฎ2๐‘Ÿ๎€ธ=โŽงโŽชโŽจโŽชโŽฉ3๐‘Ÿ24if๐‘Ÿiseven,3๐‘Ÿ2+14if๐‘Ÿisodd.(3.22)

Proof. Let ๐›ผโˆˆ{๐‘Ÿ+1,โ€ฆ,2๐‘Ÿ}, and set ๐’Ÿ๐›ผ๐‘Ÿ={(๐‘ฅ,๐‘ฆ)โˆˆ๐’ฎ2๐‘Ÿโˆถ๐‘ฅ=๐›ผ}. Therefore, Card๎€ท๐’ฎ2๐‘Ÿ๎€ธ=2๐‘Ÿ๎“๐›ผ=๐‘Ÿ+1Card๎€ท๐’Ÿ๐›ผ๐‘Ÿ๎€ธ.(3.23) Let ๐›ฝ๐›ผmax๎€ฝ=max๐›ฝโˆถ(๐›ผ,๐›ฝ)โˆˆ๐’Ÿ๐›ผ๐‘Ÿ๎€พ,๐›ฝ๐›ผmin๎€ฝ=min๐›ฝโˆถ(๐›ผ,๐›ฝ)โˆˆ๐’Ÿ๐›ผ๐‘Ÿ๎€พ.(3.24)Card(๐’Ÿ๐›ผ๐‘Ÿ)=๐›ฝ๐›ผmaxโˆ’๐›ฝ๐›ผmin+1.
It is easy to prove that Card๎€ท๐’Ÿ๐›ผ๐‘Ÿ๎€ธ๎‚ž๐›ผ=(3๐‘Ÿ+1)โˆ’2๐›ผ+2๎‚Ÿ=โŽงโŽชโŽจโŽชโŽฉ6๐‘Ÿ+2โˆ’3๐›ผ2if๐›ผiseven,6๐‘Ÿ+1โˆ’3๐›ผ2if๐›ผisodd.(3.25) The result follows from the summation of all Card(๐’Ÿ๐›ผ๐‘Ÿ),โ€‰โ€‰๐›ผโˆˆ{๐‘Ÿ+1,โ€ฆ,2๐‘Ÿ}.

Theorem 3.11. dimโ„‹3(1+๐‘ )=32๐‘ 2+92๐‘ +4.(3.26)

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

Corollary 3.12. The Poincarรฉ series of the algebras โ„‹ is ๎“๐‘ƒ(โ„‹,๐‘ก)=๐‘ โ‰ฅโˆ’1๎€ทโ„‹dim3(1+๐‘ )๎€ธ๐‘ก3(๐‘ +1)=1+๐‘ก3+๐‘ก6๎€ท1โˆ’t3๎€ธ3.(3.27)

Remark 3.13. For ๐‘Ÿ=2, the case of Figure 1, our formula gives the same answer like the classical Pick's formula for integer points in a convex polygon ฮ  with integer vertices on the plane ([17, chapter 10]) Card๎€ทฮ โˆฉโ„ค2๎€ธ=Area1(ฮ )+2Card๎€ท๐œ•ฮ โˆฉโ„ค2๎€ธ+1.(3.28) Here, ๐‘ =1 and dimโ„‹3(1+1)=10. In other hand the Pick's formula ingredients are Area(1ฮ )=2||||||||||||12โˆ’44โˆ’2=6,2Card๎€ท๐œ•ฮ โˆฉโ„ค2๎€ธ=3,(3.29) hence 6+3+1=10.

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

4. ๐ป-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

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 ๐œ†=(๐œ†0,๐œ†2,โ€ฆ,๐œ†๐‘›โˆ’1) of integers to be |๐œ†|=๐œ†0+โ‹ฏ+๐œ†๐‘›โˆ’1. If sequence ๐œ† of weight ๐‘ has all parts nonnegative, it is called a composition of ๐‘; if, in addition, ๐œ† is a nonincreasing sequence, we call it a partition of ๐‘.

Given an ๐‘Ÿร—๐‘› integer matrix ๐ถ=[๐‘๐‘–,๐‘—],โ€‰โ€‰(๐‘–,๐‘—)โˆˆ({โˆ’1}โˆชโ„ค/๐‘Ÿโ„ค)ร—โ„ค/๐‘›โ„ค, consider the set ๐‘†๐ถ of nonnegative integer sequences ๐œ†=(๐œ†1,๐œ†2,โ€ฆ,๐œ†๐‘›) satisfying the constraints๐‘๐‘–,โˆ’1+๐‘๐‘–,0๐œ†0+๐‘๐‘–,1๐œ†1+โ‹ฏ+๐‘๐‘–,๐‘›โˆ’1๐œ†๐‘›โˆ’1โ‰ง0,0โ‰ค๐‘Ÿโ‰ค๐‘›โˆ’1.(4.1) The associated full generating function is defined as follows:๐น๐ถ๎€ท๐‘ฅ0,๐‘ฅ2,โ€ฆ,๐‘ฅ๐‘›โˆ’1๎€ธ=๎“๐œ†โˆˆ๐‘†๐ถ๐‘ฅ๐œ†00๐‘ฅ๐œ†11โ‹ฏ๐‘ฅ๐œ†๐‘›โˆ’1๐‘›โˆ’1.(4.2) This function โ€œencapsulatesโ€ the solution set ๐‘†๐ถ: the coefficient of ๐‘ž๐‘ in ๐น๐ถ(๐‘ž๐‘ฅ0, ๐‘ž๐‘ฅ1,โ€ฆ,๐‘ž๐‘ฅ๐‘›โˆ’1) is a โ€œlistingโ€ (as the terms of a polynomial) of all nonnegative integer solutions to (4.1) of weight ๐‘, and the number of such solutions is the coefficient of ๐‘ž๐‘ in ๐น๐ถ(๐‘ž,๐‘ž,โ€ฆ,๐‘ž).

4.2. Formulation of the Problem in Any Dimension

Let โ„›=โ„‚[๐‘ฅ0,๐‘ฅ1,โ€ฆ,๐‘ฅ๐‘›โˆ’1] be the polynomial algebra with ๐‘› generators. For given ๐‘›โˆ’2 polynomials ๐‘ƒ1,๐‘ƒ2,โ€ฆ,๐‘ƒ๐‘›โˆ’2โˆˆโ„›, one can associate the JPS ๐œ‹(๐‘ƒ1,โ€ฆ,๐‘ƒ๐‘›โˆ’2) on โ„› given by {๐‘“,๐‘”}=๐‘‘๐‘“โˆง๐‘‘๐‘”โˆง๐‘‘๐‘ƒ1โˆงโ‹ฏโˆง๐‘‘๐‘ƒ๐‘›โˆ’2๐‘‘๐‘ฅ0โˆง๐‘‘๐‘ฅ1โˆงโ‹ฏโˆง๐‘‘๐‘ฅ๐‘›โˆ’1,(4.3) for ๐‘“,๐‘”โˆˆโ„›.

We will denote by ๐‘ƒ the particular Casimir โˆ๐‘ƒ=๐‘›โˆ’2๐‘–=1๐‘ƒ๐‘– of the Poisson structure ๐œ‹(๐‘ƒ1,โ€ฆ,๐‘ƒ๐‘›โˆ’2). We suppose that each ๐‘ƒ๐‘– is homogeneous in the sense of ๐œ-degree.

Proposition 4.1. Consider a JPS ๐œ‹(๐‘ƒ1,โ€ฆ,๐‘ƒ๐‘›โˆ’2) given by homogeneous (in the sense of ๐œ-degree) polynomials ๐‘ƒ1,โ€ฆ,๐‘ƒ๐‘›โˆ’2. If ๐œ‹(๐‘ƒ1,โ€ฆ,๐‘ƒ๐‘›โˆ’2) is ๐ป-invariant, then ๎ƒฏ๐‘›๐œโˆ’๐œ›(๐œŽโ‹…๐‘ƒ)=๐œโˆ’๐œ›(๐‘ƒ)=2if๐‘›iseven,0if๐‘›isodd,(4.4) where ๐‘ƒ=๐‘ƒ1๐‘ƒ2โ‹ฏ๐‘ƒ๐‘›โˆ’2.

Proof. Let ๐‘–<๐‘—โˆˆโ„ค/๐‘›โ„ค, and consider the set ๐ผ๐‘–,๐‘—, formed by the integers ๐‘–1<๐‘–2<โ‹ฏ<๐‘–๐‘›โˆ’2โˆˆโ„ค/๐‘›โ„คโงต{๐‘–,๐‘—}. We denote by ๐‘†๐‘–,๐‘— the set of all permutation of elements of ๐ผ๐‘–,๐‘—. We have ๎€ฝ๐‘ฅ๐‘–,๐‘ฅ๐‘—๎€พ=(โˆ’1)๐‘–+๐‘—โˆ’1๐‘‘๐‘ฅ๐‘–โˆง๐‘‘๐‘ฅ๐‘—โˆง๐‘‘๐‘ƒ1โˆง๐‘‘๐‘ƒ2โˆงโ‹ฏโˆง๐‘‘๐‘ƒ๐‘›โˆ’2๐‘‘๐‘ฅ0โˆง๐‘‘๐‘ฅ1โˆง๐‘‘๐‘ƒ2โˆงโ‹ฏโˆง๐‘‘๐‘ฅ๐‘›โˆ’1=(โˆ’1)๐‘–+๐‘—โˆ’1โˆ‘๐›ผโˆˆ๐‘†๐‘–,๐‘—(โˆ’1)|๐›ผ|๐œ•๐‘ƒ1๐œ•๐‘ฅ๐›ผ(๐‘–1)โ‹ฏ๐œ•๐‘ƒ๐‘›โˆ’2๐œ•๐‘ฅ๐›ผ(๐‘–๐‘›โˆ’2).(4.5) From the ๐œ-degree condition, ๎€ท๎€ท๐‘ƒ๐‘–+๐‘—โ‰ก๐œโˆ’๐œ›1๎€ธ๎€ท๐‘–โˆ’๐›ผ1๎€ท๎€ท๐‘ƒ๎€ธ๎€ธ+โ‹ฏ+๐œโˆ’๐œ›๐‘›โˆ’2๎€ธ๎€ท๐‘–โˆ’๐›ผ๐‘›โˆ’2๎€ธ๎€ธmodulo๐‘›.(4.6)
We can deduce, therefore, that ๎€ท๐‘ƒ๐œโˆ’๐œ›1โ‹ฏ๐‘ƒ๐‘›โˆ’2๎€ธโ‰ก๐‘›(๐‘›โˆ’1)2modulo๐‘›.(4.7) And we obtain the first part of the result. The second part is the direct consequence of facts that ๎€ฝ๐‘ฅ๐œŽโ‹…๐‘–,๐‘ฅ๐‘—๎€พ=๎€ฝ๐‘ฅ๐‘–+1,๐‘ฅ๐‘—+1๎€พ=(โˆ’1)๐‘–+๐‘—โˆ’1๎“๐›ผโˆˆ๐‘†๐‘–,๐‘—(โˆ’1)|๐›ผ|๐œ•๎€ท๐œŽโ‹…๐‘ƒ1๎€ธ๐œ•๐‘ฅ๐›ผ(๐‘–1)+1โ‹ฏ๐œ•๎€ท๐œŽโ‹…๐‘ƒ๐‘›โˆ’2๎€ธ๐œ•๐‘ฅ๐›ผ(๐‘–๐‘›โˆ’2)+1,(4.8)๐›ผ(๐‘–1)+1โ‰ โ‹ฏโ‰ ๐›ผ(๐‘–๐‘›โˆ’2)+1โˆˆ๐‘/๐‘›โ„คโงต{๐‘–+1,๐‘—+1} and the ๐œ-degree condition.

Set ๎ƒฏ๐‘›๐‘™=2if๐‘›iseven,0if๐‘›isodd.(4.9) Let โ„‹ be the set of all ๐‘„โˆˆโ„› such that ๐œโˆ’๐œ›(๐œŽโ‹…๐‘„)=๐œโˆ’๐œ›(๐‘„)=๐‘™. One can easily check the following result.

Proposition 4.2. โ„‹ is a subvector space of โ„›. It is subalgebra of โ„› if ๐‘™=0.

We endow โ„‹ with the usual grading of the polynomial algebra โ„›. For ๐‘„, an element of โ„›, we denote by ๐œ›(๐‘„) its usual weight degree. We denote by โ„‹๐‘– the homogeneous subspace of โ„‹ of degree ๐‘–.

Proposition 4.3. If ๐‘› is not a divisor of ๐‘– (in other words, ๐‘–โ‰ ๐‘›๐‘š) then โ„‹๐‘–=0.

Proof. It is clear the โ„‹0=โ„‚. We suppose now that ๐‘–โ‰ 0. Let ๐‘„โˆˆโ„‹๐‘–,โ€‰โ€‰๐‘„โ‰ 0. Then, ๎“๐‘„=๐‘˜1,โ€ฆ,๐‘˜๐‘–โˆ’1๐‘Ž๐‘˜1,โ€ฆ,๐‘˜๐‘–โˆ’1๐‘ฅ๐‘˜1โ‹ฏ๐‘ฅ๐‘˜๐‘–โˆ’1๐‘ฅ๐‘™โˆ’๐‘˜1โˆ’โ‹ฏโˆ’๐‘˜๐‘–โˆ’1.(4.10) Hence, ๎“๐œŽโ‹…๐‘„=๐‘˜1,โ€ฆ,๐‘˜๐‘–โˆ’1๐‘Ž๐‘˜1,โ€ฆ,๐‘˜๐‘–โˆ’1๐‘ฅ๐‘˜1+1โ‹ฏ๐‘ฅ๐‘˜๐‘–โˆ’1+1๐‘ฅ๐‘™โˆ’๐‘˜1โˆ’โ‹ฏโˆ’๐‘˜๐‘–โˆ’1+1.(4.11) Since ๐œโˆ’๐œ›(๐œŽโ‹…๐‘„)=๐œโˆ’๐œ›(๐‘„)=๐‘™,โ€‰โ€‰๐‘–โ‰ก0modulo๐‘›.

Set โˆ‘๐‘„=๐›ฝ๐‘ฅ๐›ผ00๐‘ฅ๐›ผ11โ‹ฏ๐‘ฅ๐›ผ๐‘›โˆ’1๐‘›โˆ’1. We suppose that ๐œ›(๐‘„)=๐‘›(1+๐‘ ). We want to find all ๐›ผ0,๐›ผ1,โ€ฆ,๐›ผ๐‘›โˆ’1 such that ๐‘„โˆˆโ„‹ and, therefore, the dimension โ„‹3(1+๐‘ ) as โ„‚-vector space.

Proposition 4.4. There exist ๐‘ 0,๐‘ 1,โ€ฆ,๐‘ ๐‘›โˆ’1 such that ๐›ผ0+๐›ผ1+โ‹ฏ+๐›ผ๐‘›โˆ’1=๐‘›(1+๐‘ ),0๐›ผ0+๐›ผ1+2๐›ผ2+โ‹ฏ(๐‘›โˆ’1)๐›ผ๐‘›โˆ’1=๐‘™+๐‘›๐‘ 0,1๐›ผ0+2๐›ผ1+3๐›ผ2+โ‹ฏ(๐‘›โˆ’1)๐›ผ๐‘›โˆ’2+0๐›ผ๐‘›โˆ’1=๐‘™+๐‘›๐‘ 1,โ‹ฎ(๐‘›โˆ’2)๐›ผ0+(๐‘›โˆ’1)๐›ผ1+0๐›ผ2+โ‹ฏ(๐‘›โˆ’3)๐›ผ๐‘›โˆ’4+(๐‘›โˆ’3)๐›ผ๐‘›โˆ’1=๐‘™+๐‘›๐‘ ๐‘›โˆ’2,(๐‘›โˆ’1)๐›ผ0+0๐›ผ1+1๐›ผ2+โ‹ฏ(๐‘›โˆ’3)๐›ผ๐‘›โˆ’3+(๐‘›โˆ’2)๐›ผ๐‘›โˆ’1=๐‘™+n๐‘ ๐‘›โˆ’1.(4.12)

Proof. That is, the direct consequence of the fact that ๐œโˆ’๐œ›(๐œŽโ‹…๐‘„)=๐œโˆ’๐œ›(๐‘„)=๐‘™.

One can easily obtain the following result.

Proposition 4.5. The system equation (4.12) has as a solution ๐›ผ๐‘–=๐‘ ๐‘›โˆ’๐‘–โˆ’1โˆ’๐‘ ๐‘›โˆ’๐‘–โ„ค+๐‘Ÿ,๐‘–โˆˆ,๐‘›โ„ค(4.13) where ๐‘Ÿ=๐‘ +1 and the ๐‘ 0,โ€ฆ,๐‘ ๐‘›โˆ’1 satisfy the condition ๐‘ 0+๐‘ 1+โ‹ฏ๐‘ ๐‘›โˆ’1=(๐‘›โˆ’1)๐‘›2๐‘Ÿโˆ’๐‘™.(4.14)

Therefore ๐›ผ0,๐›ผ1,โ€ฆ,๐›ผ๐‘›โˆ’1 are completely determined by the set of nonnegative integer sequences (๐‘ 0,๐‘ 1,โ€ฆ,๐‘ ๐‘›โˆ’1) satisfying the constraints๐‘๐‘–โˆถ๐‘ ๐‘›โˆ’๐‘–โˆ’1โˆ’๐‘ ๐‘›โˆ’๐‘–โ„ค+๐‘Ÿโ‰ฅ0,๐‘–โˆˆ,๐‘›โ„ค(4.15) and such that๐‘ 0+๐‘ 1+โ‹ฏ+๐‘ ๐‘›โˆ’1=(๐‘›โˆ’1)๐‘›2๐‘Ÿโˆ’๐‘™.(4.16) There are two approaches to determine the dimension of โ„‹๐‘›๐‘Ÿ.

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

๐‘ ๐‘›โˆ’1๎€ท๐‘ =โˆ’0+๐‘ 1+โ‹ฏ+๐‘ ๐‘›โˆ’2๎€ธ+(๐‘›โˆ’1)๐‘›2๐‘Ÿโˆ’๐‘™.(4.17)

Therefore, by replacing ๐‘ ๐‘›โˆ’1 by this value, ๐›ผ0,โ€ฆ,๐›ผ๐‘›โˆ’1 are completely determined by the set of nonnegative integer sequences (๐‘ 0,๐‘ 1,โ€ฆ,๐‘ ๐‘›โˆ’2) satisfying the constraints๐‘๎…žโˆ’1โˆถ๐‘ 0+๐‘ 1+โ‹ฏ+๐‘ ๐‘›โˆ’3+๐‘ ๐‘›โˆ’2โ‰ค(๐‘›โˆ’1)๐‘›2๐‘๐‘Ÿโˆ’๐‘™,๎…ž0โˆถ2๐‘ 0+๐‘ 1+โ‹ฏ+๐‘ ๐‘›โˆ’3+๐‘ ๐‘›โˆ’2โ‰ค๎‚ธ(๐‘›โˆ’1)๐‘›2๎‚น๐‘+1๐‘Ÿโˆ’๐‘™,๎…ž1โˆถ๐‘ 0+๐‘ 1+โ‹ฏ+๐‘ ๐‘›โˆ’3+2+๐‘ ๐‘›โˆ’2โ‰ฅ๎‚ธ(๐‘›โˆ’1)๐‘›2๎‚น๐‘๐‘˜+1๐‘Ÿโˆ’๐‘™,๎…ž๐‘–โˆถ๐‘ ๐‘›โˆ’๐‘–โˆ’1โˆ’๐‘ ๐‘›โˆ’๐‘–+๐‘Ÿโ‰ฅ0,๐‘–โˆˆโ„ค/๐‘›โ„คโงต{0,1}.(4.18) Hence, the dimension โ„‹๐‘›๐‘Ÿ is just the number of nonnegative integer points contained in the polytope given by the system (4.18), where ๐‘Ÿ=๐‘ +1.

In dimension 3, one obtains the triangle in โ„2 given by the vertices ๐ด(0,2๐‘Ÿ), ๐ต(๐‘Ÿ,2๐‘Ÿ),โ€‰โ€‰ and ๐ถ(2๐‘Ÿ,0) (see Section 3).

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

252186.fig.002
Figure 2: An example of the polytope ๐‘‡4 in the case ๐‘Ÿ=4. The vertices are in (โˆ’1/2,๐‘Ÿโˆ’1/2,2๐‘Ÿโˆ’1/2), (๐‘Ÿโˆ’2/3,2๐‘Ÿโˆ’2/3,3๐‘Ÿโˆ’2/3), (๐‘Ÿ,2๐‘Ÿโˆ’1,3๐‘Ÿโˆ’1), (๐‘Ÿ,2๐‘Ÿ,3๐‘Ÿโˆ’2), (2๐‘Ÿโˆ’1/2,3๐‘Ÿโˆ’1/2,โˆ’1/2), and (3๐‘Ÿโˆ’1/2,โˆ’1/2,๐‘Ÿโˆ’1/2).

For the second method, one can observe that the dimension of โ„‹๐‘›๐‘Ÿ is nothing else that the cardinality of the set ๐‘†๐ถ of all compositions (๐‘ 0,โ€ฆ,๐‘ ๐‘›โˆ’1) of ๐‘=((๐‘›โˆ’1)๐‘›/2)๐‘Ÿโˆ’๐‘™ subjected to the constraints (4.15). Therefore, if ๐‘†๐ถ is the set of all nonnegative integers (๐‘ 0,โ€ฆ,๐‘ ๐‘›โˆ’1) satisfying the constraints (4.15) and ๐น๐ถ is the associated generating function, then the dimension of โ„‹๐‘›๐‘Ÿ is the coefficient of ๐‘ž๐‘ in ๐น๐ถ(๐‘ž,๐‘ž,โ€ฆ,๐‘ž).The set ๐‘†๐ถ consists of all nonnegative integers points contained in the polytope of โ„๐‘›๐’ซ๐‘›โˆถ๐‘ฅ๐‘›โˆ’๐‘–โˆ’1โˆ’๐‘ฅ๐‘›โˆ’๐‘–โ„ค+๐‘Ÿโ‰ฅ0,๐‘–โˆˆ๐‘ฅ๐‘›โ„ค๐‘–โ„คโ‰ฅ0,๐‘–โˆˆ.๐‘›โ„ค(4.19)

(See Figure 3).

252186.fig.003
Figure 3: An example of the polytope ๐’ซ3 in the case ๐‘Ÿ=2.

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 โ€œFond National de Recherche (Luxembourg)โ€. 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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