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

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 𝑛 variables ℂ[𝑥₀,𝑥₁,…,𝑥_𝑛−1] over ℂ and the action of some subgroup 𝐻_𝑛 of 𝐺𝐿_𝑛(ℂ) generated by the shifts operators 𝑥_𝑖→𝑥_𝑖+1(modℤ_𝑛) and by the operators 𝑥→𝜀^𝑖𝑥_𝑖, where 𝜀^𝑛=1. We are interested in the polynomial Poisson brackets on ℂ[𝑥₀,𝑥₁,…,𝑥_𝑛−1] 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 𝑛 variables. The above-mentioned Heisenberg group action provides the automorphisms of Sklyanin algebras which are compatible with the grading and defines an 𝐻_𝑛-action on the elliptic quadratic Poisson structures on ℙ^𝑛. The latter are identified with Poisson structures on some moduli spaces of the degree 𝑛 and rank 𝑘+1 vector bundles with parabolic structure (= the flag 0⊂𝐹⊂ℂ^𝑘+1 on the elliptic curve ℰ). We will denote this elliptic Poisson algebras by 𝑞_𝑛;𝑘(ℰ). The algebras 𝑞_𝑛;𝑘(𝐸) 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 {𝑃,𝑄} of two polynomials 𝑃 and 𝑄 is given by the determinant of Jacobi matrix of functions (𝑃,𝑄,𝑃₁,…,𝑃_𝑛−2). The polynomials 𝑃_𝑖, 1≤𝑖≤𝑛−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 ℂ[𝑥₀,…,𝑥_𝑛−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, 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 (𝑛=3) and Sklyanin (𝑛=4) algebras which are both elliptic and Jacobian. It is no longer true for 𝑛>4. The relations between the Sklyanin Poisson algebras 𝑞_𝑛,𝑘(ℰ) whose centrum has dimension 1 (for 𝑛 odd) and 2 (for 𝑛 even) in the case 𝑘=1 and is generated by 𝑙=𝔤𝑐𝑑(𝑛,𝑘+1) Casimirs for 𝑞_𝑛,𝑘(ℰ) for 𝑘>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 𝑛=5 was remarked in [8] and was attributed to so-called Briesckorn-Pham polynomials for 𝑛=5𝑃₁=⁴_𝑖=0𝛼_𝑖𝑥_𝑖;𝑃₂=⁴_𝑖=0𝛽_𝑖𝑥^2𝑖;𝑃₃=⁴_𝑖=0𝛾_𝑖𝑥^2𝑖.(1.1) It is easy to check that the homogeneous quintic 𝑃=𝑃₁𝑃₂𝑃₃ (see Section 4.2) defines a Casimir for some rational degeneration of (one of) elliptic algebras 𝑞_5,1(ℰ) and 𝑞_5,2(ℰ) if it satisfies the 𝐻-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 ℋ of polynomial algebra. This vector space is completely determined by some enumerative problem of a number-theoretic type. More precisely, the homogeneous subspace ℋ_𝑖 of ℋ of degree 𝑖 is in bijection with integer solutions of a system of Diophant inequalities. Geometric interpretation of the dimension of ℋ_𝑖 is described in terms of integer points in a convex polytope given by this Diophant system. In the special case of dimension 3, ℋ 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 𝐻-invariant not necessarily quadratic Jacobian Poisson algebras with three generators. As a byproduct, we explicitly compute the Poincaré series of ℋ. In this dimension, we observe that the 𝐻-invariant JPS of degree 5 is given by the Casimir sextic 𝑃^∨=16𝑎𝑥⁶⁰+𝑥⁶¹+𝑥⁶²+13𝑏𝑥³⁰+1𝑥³¹+𝑥³⁰𝑥³²+𝑥³¹𝑥³²+𝑐𝑥⁴⁰𝑥₁𝑥₂+𝑥₀𝑥⁴¹𝑥₂+𝑥₀𝑥₁𝑥⁴²2𝑑𝑥²⁰𝑥²¹𝑥²²,(1.2)𝑎,𝑏,𝑐,𝑑∈ℂ. This structure is a “projectively dual” to the Artin-Schelter-Tate elliptic Poisson structure which is the 𝐻-invariant JPS given by the cubic 𝑃=𝑥³⁰+𝑥³¹+𝑥³²+𝛾𝑥₀𝑥₁𝑥₂,(1.3) where 𝛾∈ℂ. In fact, the algebraic variety ℰ^∨∶{𝑃^∨=0}∈ℙ² is the (generically) projectively dual to the elliptic curve ℰ∶{𝑃=0}⊂ℙ².

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, 𝐾 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 ℛ be a commutative 𝐾-algebra. One says that ℛ is a Poisson algebra if ℛ is endowed with a Lie bracket, indicated with {⋅,⋅}, which is also a biderivation. One can also say that ℛ is endowed with a Poisson structure, and therefore, the bracket {⋅,⋅} is called the Poisson bracket. Elements of the center are called Casimirs: 𝑎∈ℛ is a Casimir if {𝑎,𝑏}=0 for all 𝑏∈ℛ.

A Poisson manifold 𝑀 (smooth, algebraic, etc.) is a manifold whose function algebra 𝒜 (𝐶^∞(𝑀), 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 𝑞₁=12𝑥²⁰𝑞₂+𝑥²²+𝑘𝑥₁𝑥₃,=12𝑥²¹+𝑥²³+𝑘𝑥₀𝑥₂(2.1) be two elements of ℂ[𝑥₀,𝑥₁,𝑥₂,𝑥₃], where 𝑘∈ℂ.

On ℂ[𝑥₀,𝑥₁,𝑥₂,𝑥₃], a Poisson structure 𝜋 is defined by {𝑓,𝑔}_𝜋∶=𝑑𝑓∧𝑑𝑔∧𝑑𝑞¹∧𝑑𝑞₂𝑑𝑥₀∧𝑑𝑥₁∧𝑑𝑥₂∧𝑑𝑥₃,(2.2) or more explicitly (mod ℤ₄) 𝑥_𝑖𝑥_𝑖,𝑥_𝑖+1=𝑘²𝑥_𝑖𝑥_𝑖+1−𝑥_𝑖+2𝑥_𝑖+3,,𝑥_𝑖+2=𝑘𝑥^2𝑖+3−𝑥^2𝑖+1,𝑖=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 “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 ℰ given by two quadrics 𝑞_1,2. We can think about this curve ℰ as a complete intersection of the couple 𝑞₁=0, 𝑞₂=0 embedded in 𝒞𝑃³ (as it was observed in Sklyanin's initial paper).

A possible generalization one can be obtained considering 𝑛−2 polynomials 𝑄_𝑖 in 𝐾^𝑛 with coordinates 𝑥_𝑖, 𝑖=0,…,𝑛−1. We can define a bilinear differential operation {⋅,⋅}∶𝐾𝑥₁,…,𝑥_𝑛⊗𝐾𝑥₁,…,𝑥_𝑛⟶𝐾𝑥₁,…,𝑥_𝑛,(2.4) by{𝑓,𝑔}=𝑑𝑓∧𝑑𝑔∧𝑑𝑄¹∧⋯∧𝑑𝑄_𝑛−2𝑑𝑥₁∧𝑑𝑥₂∧⋯∧𝑑𝑥_𝑛,𝑓,𝑔∈𝐾𝑥₁,…,𝑥_𝑛.(2.5) This operation, which gives a Poisson algebra structure on 𝐾[𝑥₁,…,𝑥_𝑛], is called a Jacobian Poisson structure (JPS), and it is a partial case of more general 𝑛−𝑚-ary Nambu operation given by an antisymmetric 𝑛−𝑚-polyvector field introduced by Nambu [6] and was extensively studied by Takhtajan [5].

The polynomials 𝑄_𝑖,  𝑖=1,…,𝑛−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 𝑞_𝑛(ℰ,𝜂) and 𝑞_𝑛,𝑘(ℰ,𝜂)

(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 𝑄_𝑛(ℰ,𝜂) and 𝑄_𝑛,𝑘(ℰ,𝜂) [12, 13].

Let Γ=ℤ+𝜏ℤ⊂ℂ be an integral lattice generated by 1 and 𝜏∈ℂ, with Im𝜏>0. Consider the elliptic curve ℰ=ℂ/Γ and a point 𝜂 on this curve.

In their article [13], given 𝑘<𝑛, mutually prime, Odesskiĭ and Feĭgin construct an algebra, called elliptic, 𝑄_𝑛,𝑘(ℰ,𝜂), as an algebra defined by 𝑛 generators {𝑥_𝑖,𝑖∈ℤ/𝑛ℤ} and the following relations:_𝑟∈_ℤ/𝑛ℤ𝜃_{𝑗−𝑖+𝑟(𝑘−1)}(0)𝜃_𝑘𝑟(𝜂)𝜃_{𝑗−𝑖−𝑟}(−𝜂)𝑥^𝑗−𝑟𝑥_𝑖+𝑟=0,𝑖≠𝑗,𝑖,𝑗∈ℤ𝑛ℤ,(2.6) where 𝜃_𝛼 are theta functions [13].

These family of algebras has the following properties: (1)the center of the algebra 𝑄_𝑛,𝑘(ℰ,𝜂), for generic ℰ and 𝜂, is the algebra of polynomial of 𝑚=𝑝𝑔𝑐𝑑(𝑛,𝑘+1) variables of degree 𝑛/𝑚,(2)𝑄_𝑛,𝑘(ℰ,0)=ℂ[𝑥₁,…,𝑥_𝑛] is commutative, (3)𝑄_{𝑛,𝑛−1}(ℰ,𝜂)=ℂ[𝑥₁,…,𝑥_𝑛] is commutative for all 𝜂, (4)𝑄_𝑛,𝑘(ℰ,𝜂)≃𝑄_𝑛,𝑘^′(ℰ,𝜂), if 𝑘𝑘^≡1 (mod 𝑛), (5)the maps 𝑥_𝑖↦𝑥_𝑖+1 et 𝑥_𝑖↦𝜀^𝑖𝑥_𝑖, where 𝜀^𝑛=1, define automorphisms of the algebra 𝑄_𝑛,𝑘(ℰ,𝜂),(6)the algebras 𝑄_𝑛,𝑘(ℰ,𝜂) are deformations of polynomial algebras. The associated Poisson structure is denoted by 𝑞_𝑛,𝑘(ℰ,𝜂), (7)among the algebras 𝑞_𝑛,𝑘(ℰ,𝜂), only 𝑞₃(ℰ,𝜂) (the Artin-Schelter-Tate algebra) and the Sklyanin algebra 𝑞₄(ℰ,𝜂) are Jacobian Poisson structures.

2.3. The Heisenberg Invariant Poisson Structures

2.3.1. The G-Invariant Poisson Structures

Let 𝐺 be a group acting on a Poisson algebra ℛ.

Definition 2.1. A Poisson bracket {⋅,⋅} on ℛ is said to be a 𝐺-invariant if 𝐺 acts on ℛ by Poisson automorphisms.
In other words, for every 𝑔∈𝐺, the morphism 𝜑_𝑔∶ℛ→ℛ, 𝑎↦𝑔⋅𝑎 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 𝐻-invariant Poisson structures. That is, a special case of a 𝐺-invariant structure when 𝐺 in the finite Heisenberg group and ℛ is the polynomial algebra. Let us remember this notion.

Let 𝑉 be a complex vector space of dimension 𝑛 and 𝑒₀,…,𝑒_𝑛−1 a basis of 𝑉. Take the 𝑛th primitive root of unity 𝜀=𝑒^{2𝜋𝑖/𝑛}.

Consider 𝜎,𝜏 of 𝐺𝐿(𝑉) defined by 𝜏𝜎𝑒𝑒_𝑚𝑚=𝑒_𝑚−1,=𝜀^𝑚𝑒_𝑚.(2.8) The Heisenberg of dimension 𝑛 is nothing else that the subspace 𝐻_𝑛⊂𝐺𝐿(𝑉) generated by 𝜎 and 𝜏.

From now on, we assume that 𝑉=ℂ^𝑛, with 𝑥₀,𝑥₁,…,𝑥_𝑛−1 as a basis and consider the coordinate ring ℂ[𝑥₀,𝑥₁,…,𝑥_𝑛−1].

Naturally, 𝜎 and 𝜏 act by automorphisms on the algebra ℂ[𝑥₀,𝑥₁,…,𝑥_𝑛−1] as follows: 𝜏⋅𝛼𝑥_𝛼00𝜎⋅𝛼𝑥^𝛼0₀𝑥^𝛼1₁⋯𝑥^𝛼𝑛−1_𝑛−1=𝛼𝑥^𝛼𝑛−1₀𝑥^𝛼0₁⋯𝑥^𝛼𝑛−2_𝑛−1,𝑥^𝛼1₁⋯𝑥^𝛼𝑛−1_𝑛−1=𝜀^{𝛼1+2𝛼2+⋯+(𝑛−1)𝛼𝑛−1}𝛼𝑥^𝛼0₀𝑥^𝛼1₁⋯𝑥^𝛼𝑛−1_𝑛−1.(2.9) We introduced in [2] the notion of 𝜏-degree on the polynomial algebra ℂ[𝑥₀,𝑥₁,…,𝑥_𝑛−1]. The 𝜏-degree of a monomial 𝑀=𝛼𝑥^𝛼0₀𝑥^𝛼1₁⋯𝑥^𝛼𝑛−1_𝑛−1 is the positive integer 𝛼₁+2𝛼₂+⋯+(𝑛−1)𝛼_𝑛−1∈ℤ/𝑛ℤ if 𝛼≠0 and −∞ if not. The 𝜏-degree of 𝑀 is denoted 𝜏𝜛(𝑀). A 𝜏-degree of a polynomial is the highest 𝜏-degree of its monomials.

For simplicity, the 𝐻_𝑛-invariance condition will be referred from now on just as 𝐻-invariance. An 𝐻-invariant Poisson bracket on 𝒜=ℂ[𝑥₀,𝑥₁,…,𝑥_𝑛−1] is nothing but a bracket on 𝒜 which satisfy the following: 𝜏𝑥⋅𝑥_𝑖+1,_𝑖𝑥_𝑗+1=𝜎⋅𝑥_𝑖,𝑥_𝑗,,𝑥_𝑗=𝜀^𝑖+𝑗𝑥_𝑖,𝑥_𝑗,(2.10) for all 𝑖,𝑗∈ℤ/𝑛ℤ.

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

Proposition 2.2 (see [2]). The Sklyanin-Odesskii-Feigin Poisson algebras 𝑞_𝑛,𝑘(ℰ) are 𝐻-invariant Poisson algebras.

Therefore, an 𝐻-invariant Poisson structures on the polynomial algebra ℛ 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 𝐻-invariant Poisson structures.

Proposition 2.3 (see [2]). If {⋅,⋅} is an 𝐻-invariant polynomial Poisson bracket, the usual polynomial degree of the monomial of {𝑥_𝑖,𝑥_𝑗} equals to 2+𝑠𝑛,  𝑠∈ℕ.

Proposition 2.4 (see [2]). Let 𝑃∈ℛ=ℂ[𝑥₀,…,𝑥_𝑛−1].
For all 𝑖∈{0,…,𝑛−1}, 𝜎⋅𝜕𝐹𝜕𝑥_𝑖=𝜕(𝜎⋅𝐹)𝜕𝜎⋅𝑥_𝑖.(2.11)

3. 𝐻-Invariant JPS in Dimension 3

We consider first a generalization of Artin-Schelter-Tate quadratic Poisson algebras. Let ℛ=ℂ[𝑥₀,𝑥₁,𝑥₂] be the polynomial algebra with 3 generators. For every 𝑃∈𝒜, we have a JPS 𝜋(𝑃) on ℛ given by 𝑥_𝑖,𝑥_𝑗=𝜕𝜕𝑥𝑃_𝑘,(3.1) where (𝑖,𝑗,𝑘)∈ℤ/3ℤ is a cyclic permutation of (0,1,2). Let ℋ be the set of all 𝑃∈𝒜 such that 𝜋(𝑃) is an 𝐻-invariant Poisson structure.

Proposition 3.1. If 𝑃∈ℋ is a homogeneous polynomial, then 𝜎⋅𝑃=𝑃 and 𝜏𝜛(𝑃)=0.

Proof. Let (𝑖,𝑗,𝑘)∈(ℤ/3ℤ)³ be a cyclic permutation of (0,1,2). One has 𝑥_𝑖,𝑥_𝑗=𝜕𝜕𝑥𝑃_𝑘,(3.2)𝑥_𝑖+1,𝑥_𝑗+1=𝜕𝑥𝜕𝑃_𝑘+1.(3.3) Using Proposition 2.4, we conclude that for all 𝑚∈ℤ/3ℤ,  𝜕𝜎𝑃/𝜕𝑥_𝑚=𝜕𝑃/𝜕𝑥_𝑚.
It gives that 𝜎⋅𝑃=𝑃.
On the other hand, from (3.2), one has 𝜏−𝜛(𝑃)≡𝑖+𝑗+𝑘mod3. And we get the second half of the proposition.

Proposition 3.2. ℋ is a subalgebra of ℛ.

Proof. Let 𝑃,𝑄∈ℋ. It is clear that for all 𝛼,𝛽∈ℂ,  𝛼𝑃+𝛽𝑄 belongs to ℋ.
Let us denote by {⋅,⋅}_𝐹 the JPS associated with the polynomial 𝐹∈ℛ. It is easy to verify that {𝑥_𝑖,𝑥_𝑗}_𝑃𝑄=𝑃{𝑥_𝑖,𝑥_𝑗}_𝑄+𝑄{𝑥_𝑖,𝑥_𝑗}_𝑃. Therefore, it is clear that the 𝐻-invariance condition is verified for the JPS associated to the polynomial 𝑃𝑄.

We endow ℋ with the usual grading of the polynomial algebra ℛ. For 𝐹, an element of ℛ, we denote by 𝜛(𝐹) its usual weight degree. We denote by ℋ_𝑖 the homogeneous subspace of ℋ of degree 𝑖.

Proposition 3.3. If 3 does not divide 𝑖∈ℕ (in other words 𝑖≠3𝑘), then ℋ_𝑖=0.

Proof. First of all ℋ₀=ℂ. We suppose now that 𝑖≠0. Let 𝑃∈ℋ_𝑖,𝑃≠0. Then, 𝜛(𝑃)=𝑖. It follows from Proposition 2.3 and the definition of the Poisson brackets that there exists 𝑠∈ℕ such that 𝜛(𝑥_𝑖,𝑥_𝑗)=2+3𝑠. The result follows from the fact that 𝜛(𝑥_𝑖,𝑥_𝑗)=𝜛(𝑃)−1.

Set 𝑃=∑𝑐_𝛼𝑥^𝛼0₀𝑥^𝛼1₁𝑥^𝛼2₂, where 𝛼=(𝛼₀,𝛼₁,𝛼₂). We suppose that 𝜛(𝑃)=3(1+𝑠). We want to find all 𝛼₀,𝛼₁,𝛼₂ such that 𝑃∈ℋ and, therefore, the dimension ℋ_3(1+𝑠) as ℂ-vector space.

Proposition 3.4. There exist 𝑠^, 𝑠^, and 𝑠^ such that 𝛼₀0𝛼+₀𝛼₁+𝛼₂=3(1+𝑠),𝛼₀+𝛼₁+2𝛼₂=3𝑠^,2𝛼₀+2𝛼₁+0𝛼₂=3𝑠^,+0𝛼₁+1𝛼₂=3𝑠^.(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: 𝛼𝛼₀₁=4𝑟−2𝑠^−𝑠^,𝛼=₂−2𝑟+𝑠^+2𝑠^,=𝑟+𝑠^−𝑠^,(3.5) where 𝑟=1+𝑠,  𝑠^ and 𝑠^ live in the polygon given by the following inequalities in ℝ²∶−𝑥2−𝑥𝑥−𝑥++2+𝑦𝑦𝑦𝑦⩽⩽⩽⩽34−𝑟𝑟𝑟,.2,𝑟,(3.6)

Remark 3.6. For 𝑟=1, one obtains the Artin-Schelter-Tate Poisson algebra which is the JPS given by the Casimir 𝑃=𝛼(𝑥³⁰+𝑥³¹+𝑥³²)+𝛽𝑥₀𝑥₁𝑥₂,  𝛼,𝛽∈ℂ. Suppose that 𝛼≠0, then it can take the form 𝑃=𝑥³⁰+𝑥³¹+𝑥³²+𝛾𝑥₀𝑥₁𝑥₂,(3.7) where 𝛾∈ℂ. The interesting feature of this Poisson algebra is that their polynomial character is preserved even after the following nonalgebraic changes of variables. Let 𝑦₀=𝑥₀;𝑦₁=𝑥₁𝑥^2−1/2;𝑦₂=𝑥^23/2.(3.8) The polynomial 𝑃 in the coordinates (𝑦₀;𝑦₁;𝑦₂) has the form 𝑃=𝑦³⁰+𝑦³¹𝑦₂+𝑦²²+𝛾𝑦₀𝑦₁𝑦₂.(3.9) The Poisson bracket is also polynomial (which is not evident at all!) and has the same form 𝑥_𝑖,𝑥_𝑗=𝜕𝜕𝑥𝑃_𝑘,(3.10) where (𝑖,𝑗,𝑘) 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: (ℂ^∗)³×ℙ²→ℙ given by 𝜆⋅𝑥₀∶𝑥₁∶𝑥₂=𝜆²𝑥₀∶𝜆𝑥₁∶𝜆³𝑥₂.(3.11)
Put deg𝑦₀=2;deg𝑦₂=1;deg𝑦₂=3. Then, the polynomial 𝑃 is also homogeneous in (𝑦₀;𝑦₁;𝑦₂) and defines an elliptic curve 𝑃=0 in the weighted projective space 𝕎𝑃_2;1;3.
The similar change of variables 𝑧₀=𝑥^0−3/4𝑥^13/2;𝑧₁=𝑥^01/4𝑥^1−1/2𝑥₂;𝑧₂=𝑥^03/2(3.12) defines the JPS structure invariant with respect to the torus action (ℂ^∗)³×ℙ²→ℙ given by 𝜆⋅𝑥₀∶𝑥₁∶𝑥₂=𝜆𝑥₀∶𝜆𝑥₁∶𝜆²𝑥₂,(3.13) and related to the elliptic curve 1/3(𝑧²²+𝑧²⁰𝑧₂+𝑧₀𝑧³¹)+𝑘𝑧₀𝑧₁𝑧₂=0 in the weighted projective space 𝕎𝑃_1;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 ℝ² given by the system (3.6) is a triangle 𝒯_𝑟 with (0,𝑟),  (𝑟,2𝑟), and (2𝑟,0) as vertices. Then, dimℋ_3𝑟=Card(𝒯_𝑟∩ℕ²).

Remark 3.8. For 𝑟=2, the case of Figure 1, the generic Heisenberg-invariant JPS is given by the sextic polynomial 𝑃^∨=16𝑎𝑝⁶⁰+𝑝⁶¹+𝑝⁶²+13𝑏𝑝³⁰𝑝³¹+𝑝³⁰𝑝³²+𝑝³¹𝑝³²+𝑐𝑝⁴⁰𝑝₁𝑝₂+𝑝₀𝑝⁴¹𝑝₂+𝑝₀𝑝₁𝑝⁴²+12𝑑𝑝²⁰𝑎,𝑏,𝑐,𝑑∈𝑝²¹ℂ𝑝²².,(3.14) The corresponding Poisson bracket takes the form 𝑥_𝑖,𝑥_𝑗=𝑎𝑥^5𝑘+𝑏𝑥^3𝑖+𝑥^3𝑗𝑥^2𝑘+𝑐𝑥_𝑖𝑥_𝑗𝑥^3𝑖+𝑥^3𝑗+4𝑥^3𝑘+𝑑𝑥^2𝑖𝑥^2𝑗𝑥_𝑘,(3.15) where 𝑖,𝑗,𝑘 are the cyclic permutations of 0,1,2.
This new JPS should be considered as the “projectively dual” to the Artin-Schelter-Tate JPS, since the algebraic variety ℰ^∨∶𝑃^∨=0 is generically the projective dual curve in ℙ² to the elliptic curve ℰ∶𝑃=𝑥³⁰+𝑥³¹+𝑥³²+𝛾𝑥₀𝑥₁𝑥₂=0.(3.16)
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 ℰ=0⊂ℙ². The coordinates (𝑝₀∶𝑝₁∶𝑝₂)∈ℙ^2∗ of a point 𝑝∈ℙ^2∗ satisfies to the sextic relation ℰ^∨=0 if and only if the line 𝑥₀𝑝₀+𝑥₁𝑝₁+𝑥₂𝑝₂=0 is tangent to the conic locus 𝒞(𝑥,𝑝)=0, where 𝒞(𝑥,𝑝)=||||||||||||||||0𝑝⁰𝑝₁𝑝₂𝑝₀𝜕²𝑃𝜕𝑥²⁰𝜕²𝑃𝜕𝑥₀𝜕𝑥₁𝜕²𝑃𝜕𝑥₀𝜕𝑥₂𝑝₁𝜕²𝑃𝜕𝑥₁𝜕𝑥₀𝜕²𝑃𝜕𝑥₁𝜕𝑥₁𝜕²𝑃𝜕𝑥₁𝜕𝑥₂𝑝₂𝜕²𝑃𝜕𝑥₂𝜕𝑥₀𝜕²𝑃𝜕𝑥₂𝜕𝑥₁𝜕²𝑃𝜕𝑥₂𝜕𝑥₂||||||||||||||||.(3.17)
Set 𝒮_𝑟=𝒯_𝑟∩ℕ².  𝒮_𝑟=𝒮^1𝑟∪𝒮^2𝑟, 𝒮^1𝑟={(𝑥,𝑦)∈𝒮_𝑟∶0≤𝑥≤𝑟} and 𝒮^1𝑟={(𝑥,𝑦)∈𝒮_𝑟∶𝑟<𝑥≤2𝑟}.  dimℋ_3𝑟=Card(𝒮^1𝑟)+Card(𝒮^2𝑟).

Figure 1 

An example of the triangle 𝒯2 in the case 𝑟=2.

Proposition 3.9. Card𝒮^1𝑟=⎧⎪⎨⎪⎩3𝑟²+6𝑟+44if𝑟iseven,3𝑟²+6𝑟+34if𝑟isodd.(3.18)

Proof. Let 𝛼∈{0,…,𝑟}, and set 𝒟^𝛼𝑟={(𝑥,𝑦)∈𝒮^1𝑟∶𝑥=𝛼}. Therefore, Card𝒮^1𝑟=^𝑟_𝛼=0Card𝒟^𝛼𝑟.(3.19) Let 𝛽𝛽^𝛼max_𝛼min=max𝛽∶(𝛼,𝛽)∈𝒟^𝛼𝑟,=min𝛽∶(𝛼,𝛽)∈𝒟^𝛼𝑟.(3.20)Card(𝒟^𝛼𝑟)=𝛽^𝛼max−𝛽^𝛼min+1.
It is easy to prove that Card𝒟^𝛼𝑟=(𝛼+1)+𝛼2=⎧⎪⎨⎪⎩3𝛼+22if𝛼iseven,3𝛼+12if𝛼isodd.(3.21) The result follows from the summation of all Card(𝒟^𝛼𝑟),𝛼∈{0,…,𝑟}.

Proposition 3.10. Card𝒮^2𝑟=⎧⎪⎨⎪⎩34𝑟²if𝑟iseven,3𝑟²+14if𝑟isodd.(3.22)

Proof. Let 𝛼∈{𝑟+1,…,2𝑟}, and set 𝒟^𝛼𝑟={(𝑥,𝑦)∈𝒮^2𝑟∶𝑥=𝛼}. Therefore, Card𝒮^2𝑟=^2𝑟_𝛼_=𝑟+1Card𝒟^𝛼𝑟.(3.23) Let 𝛽𝛽^𝛼max_𝛼min=max𝛽∶(𝛼,𝛽)∈𝒟^𝛼𝑟,=min𝛽∶(𝛼,𝛽)∈𝒟^𝛼𝑟.(3.24)Card(𝒟^𝛼𝑟)=𝛽^𝛼max−𝛽^𝛼min+1.
It is easy to prove that Card𝒟^𝛼𝑟=(3𝑟+1)−2𝛼+𝛼2=⎧⎪⎨⎪⎩6𝑟+2−3𝛼2if𝛼iseven,6𝑟+1−3𝛼2if𝛼isodd.(3.25) The result follows from the summation of all Card(𝒟^𝛼𝑟),  𝛼∈{𝑟+1,…,2𝑟}.

Theorem 3.11. dimℋ_3(1+𝑠)=32𝑠²+92𝑠+4.(3.26)

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

Corollary 3.12. The Poincaré series of the algebras ℋ is 𝑃(ℋ,𝑡)=_𝑠≥−1dimℋ_3(1+𝑠)𝑡^3(𝑠+1)=1+𝑡³+𝑡⁶1−t³³.(3.27)

Remark 3.13. For 𝑟=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 Π with integer vertices on the plane ([17, chapter 10]) CardΠ∩ℤ²=Area(Π)+12Card𝜕Π∩ℤ²+1.(3.28) Here, 𝑠=1 and dimℋ₃₍₁₊₁₎=10. In other hand the Pick's formula ingredients are Area(Π)=12||||||24−−42||||||=6,12Card𝜕Π∩ℤ²=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. 𝐻-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 𝜆=(𝜆₀,𝜆₂,…,𝜆_𝑛−1) of integers to be |𝜆|=𝜆₀+⋯+𝜆_𝑛−1. If sequence 𝜆 of weight 𝑁 has all parts nonnegative, it is called a composition of 𝑁; if, in addition, 𝜆 is a nonincreasing sequence, we call it a partition of 𝑁.

Given an 𝑟×𝑛 integer matrix 𝐶=[𝑐_𝑖,𝑗],  (𝑖,𝑗)∈({−1}∪ℤ/𝑟ℤ)×ℤ/𝑛ℤ, consider the set 𝑆_𝐶 of nonnegative integer sequences 𝜆=(𝜆₁,𝜆₂,…,𝜆_𝑛) satisfying the constraints𝑐_𝑖,−1+𝑐_𝑖,0𝜆₀+𝑐_𝑖,1𝜆₁+⋯+𝑐_{𝑖,𝑛−1}𝜆_𝑛−1≧0,0≤𝑟≤𝑛−1.(4.1) The associated full generating function is defined as follows:𝐹_𝐶𝑥₀,𝑥₂,…,𝑥_𝑛−1=_{𝜆∈𝑆𝐶}𝑥^𝜆0₀𝑥^𝜆1₁⋯𝑥^𝜆𝑛−1_𝑛−1.(4.2) This function “encapsulates” the solution set 𝑆_𝐶: the coefficient of 𝑞^𝑁 in 𝐹_𝐶(𝑞𝑥₀, 𝑞𝑥₁,…,𝑞𝑥_𝑛−1) is a “listing” (as the terms of a polynomial) of all nonnegative integer solutions to (4.1) of weight 𝑁, and the number of such solutions is the coefficient of 𝑞^𝑁 in 𝐹_𝐶(𝑞,𝑞,…,𝑞).

4.2. Formulation of the Problem in Any Dimension

Let ℛ=ℂ[𝑥₀,𝑥₁,…,𝑥_𝑛−1] be the polynomial algebra with 𝑛 generators. For given 𝑛−2 polynomials 𝑃₁,𝑃₂,…,𝑃_𝑛−2∈ℛ, one can associate the JPS 𝜋(𝑃₁,…,𝑃_𝑛−2) on ℛ given by {𝑓,𝑔}=𝑑𝑓∧𝑑𝑔∧𝑑𝑃¹∧⋯∧𝑑𝑃_𝑛−2𝑑𝑥₀∧𝑑𝑥₁∧⋯∧𝑑𝑥_𝑛−1,(4.3) for 𝑓,𝑔∈ℛ.

We will denote by 𝑃 the particular Casimir 𝑃=∏^{𝑛−2𝑖=1}𝑃_𝑖 of the Poisson structure 𝜋(𝑃₁,…,𝑃_𝑛−2). We suppose that each 𝑃_𝑖 is homogeneous in the sense of 𝜏-degree.

Proposition 4.1. Consider a JPS 𝜋(𝑃₁,…,𝑃_𝑛−2) given by homogeneous (in the sense of 𝜏-degree) polynomials 𝑃₁,…,𝑃_𝑛−2. If 𝜋(𝑃₁,…,𝑃_𝑛−2) is 𝐻-invariant, then 𝜏−𝜛(𝜎⋅𝑃)=𝜏−𝜛(𝑃)=2𝑛if𝑛iseven,0if𝑛isodd,(4.4) where 𝑃=𝑃₁𝑃₂⋯𝑃_𝑛−2.

Proof. Let 𝑖<𝑗∈ℤ/𝑛ℤ, and consider the set 𝐼_𝑖,𝑗, formed by the integers 𝑖₁<𝑖₂<⋯<𝑖_𝑛−2∈ℤ/𝑛ℤ⧵{𝑖,𝑗}. We denote by 𝑆_𝑖,𝑗 the set of all permutation of elements of 𝐼_𝑖,𝑗. We have 𝑥_𝑖,𝑥_𝑗=(−1)^{𝑖+𝑗−1}𝑑𝑥_𝑖∧𝑑𝑥_𝑗∧𝑑𝑃₁∧𝑑𝑃₂∧⋯∧𝑑𝑃_𝑛−2𝑑𝑥₀∧𝑑𝑥₁∧𝑑𝑃₂∧⋯∧𝑑𝑥_𝑛−1=(−1)^{𝑖+𝑗−1}∑_{𝛼∈𝑆𝑖,𝑗}(−1)^|𝛼|𝜕𝑃₁𝜕𝑥_𝛼(𝑖1)⋯𝜕𝑃^𝑛−2𝜕𝑥_{𝛼(𝑖𝑛−2)}.(4.5) From the 𝜏-degree condition, 𝑖+𝑗≡𝜏−𝜛𝑃₁−𝛼𝑖₁+⋯+𝜏−𝜛𝑃_𝑛−2−𝛼𝑖_𝑛−2modulo𝑛.(4.6)
We can deduce, therefore, that 𝜏−𝜛𝑃₁⋯𝑃_𝑛−2≡𝑛(𝑛−1)2modulo𝑛.(4.7) And we obtain the first part of the result. The second part is the direct consequence of facts that 𝜎⋅𝑥_𝑖,𝑥_𝑗=𝑥_𝑖+1,𝑥_𝑗+1=(−1)^{𝑖+𝑗−1}_𝛼_{∈𝑆𝑖,𝑗}(−1)^|𝛼|𝜕𝜎⋅𝑃₁𝜕𝑥_{𝛼(𝑖1)+1}⋯𝜕𝜎⋅𝑃^𝑛−2𝜕𝑥_{𝛼(𝑖𝑛−2)+1},(4.8)𝛼(𝑖₁)+1≠⋯≠𝛼(𝑖_𝑛−2)+1∈𝑍/𝑛ℤ⧵{𝑖+1,𝑗+1} and the 𝜏-degree condition.

Set 𝑙=2𝑛if𝑛iseven,0if𝑛isodd.(4.9) Let ℋ be the set of all 𝑄∈ℛ such that 𝜏−𝜛(𝜎⋅𝑄)=𝜏−𝜛(𝑄)=𝑙. One can easily check the following result.

Proposition 4.2. ℋ is a subvector space of ℛ. It is subalgebra of ℛ if 𝑙=0.

We endow ℋ with the usual grading of the polynomial algebra ℛ. For 𝑄, an element of ℛ, we denote by 𝜛(𝑄) its usual weight degree. We denote by ℋ_𝑖 the homogeneous subspace of ℋ of degree 𝑖.

Proposition 4.3. If 𝑛 is not a divisor of 𝑖 (in other words, 𝑖≠𝑛𝑚) then ℋ_𝑖=0.

Proof. It is clear the ℋ₀=ℂ. We suppose now that 𝑖≠0. Let 𝑄∈ℋ_𝑖,  𝑄≠0. Then, 𝑄=_{𝑘1,…,𝑘𝑖−1}𝑎_{𝑘1,…,𝑘𝑖−1}𝑥_𝑘1⋯𝑥_𝑘𝑖−1𝑥_{𝑙−𝑘1−⋯−𝑘𝑖−1}.(4.10) Hence, 𝜎⋅𝑄=_{𝑘1,…,𝑘𝑖−1}𝑎_{𝑘1,…,𝑘𝑖−1}𝑥_𝑘1+1⋯𝑥_{𝑘𝑖−1+1}𝑥_{𝑙−𝑘1−⋯−𝑘𝑖−1+1}.(4.11) Since 𝜏−𝜛(𝜎⋅𝑄)=𝜏−𝜛(𝑄)=𝑙,  𝑖≡0modulo𝑛.

Set 𝑄=∑𝛽𝑥^𝛼0₀𝑥^𝛼1₁⋯𝑥^𝛼𝑛−1_𝑛−1. We suppose that 𝜛(𝑄)=𝑛(1+𝑠). We want to find all 𝛼₀,𝛼₁,…,𝛼_𝑛−1 such that 𝑄∈ℋ and, therefore, the dimension ℋ_3(1+𝑠) as ℂ-vector space.

Proposition 4.4. There exist 𝑠₀,𝑠₁,…,𝑠_𝑛−1 such that 0𝛼₀𝛼₀+𝛼₁+⋯+𝛼_𝑛−1=𝑛(1+𝑠),1𝛼₀+𝛼₁+2𝛼₂+⋯(𝑛−1)𝛼_𝑛−1=𝑙+𝑛𝑠₀,(𝑛−2)𝛼₀+2𝛼₁+3𝛼₂+⋯(𝑛−⋮1)𝛼_𝑛−2+0𝛼_𝑛−1=𝑙+𝑛𝑠₁,(𝑛−1)+𝛼₀(𝑛−1)𝛼₁+0𝛼₂+⋯(𝑛−3)𝛼_𝑛−4+(𝑛−3)𝛼_𝑛−1=𝑙+𝑛𝑠_𝑛−2,+0𝛼₁+1𝛼₂+⋯(𝑛−3)𝛼_𝑛−3+(𝑛−2)𝛼_𝑛−1=𝑙+n𝑠_𝑛−1.(4.12)

Proof. That is, the direct consequence of the fact that 𝜏−𝜛(𝜎⋅𝑄)=𝜏−𝜛(𝑄)=𝑙.

One can easily obtain the following result.

Proposition 4.5. The system equation (4.12) has as a solution 𝛼_𝑖=𝑠_{𝑛−𝑖−1}−𝑠_𝑛−𝑖+𝑟,𝑖∈ℤ𝑛ℤ,(4.13) where 𝑟=𝑠+1 and the 𝑠₀,…,𝑠_𝑛−1 satisfy the condition 𝑠₀+𝑠₁+⋯𝑠_𝑛−1=(𝑛−1)𝑛2𝑟−𝑙.(4.14)

Therefore 𝛼₀,𝛼₁,…,𝛼_𝑛−1 are completely determined by the set of nonnegative integer sequences (𝑠₀,𝑠₁,…,𝑠_𝑛−1) satisfying the constraints𝑐_𝑖∶𝑠_{𝑛−𝑖−1}−𝑠_𝑛−𝑖+𝑟≥0,𝑖∈ℤ𝑛ℤ,(4.15) and such that𝑠₀+𝑠₁+⋯+𝑠_𝑛−1=(𝑛−1)𝑛2𝑟−𝑙.(4.16) There are two approaches to determine the dimension of ℋ_𝑛𝑟.

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

𝑠_𝑛−1=−𝑠₀+𝑠₁+⋯+𝑠_𝑛−2+(𝑛−1)𝑛2𝑟−𝑙.(4.17)

Therefore, by replacing 𝑠_𝑛−1 by this value, 𝛼₀,…,𝛼_𝑛−1 are completely determined by the set of nonnegative integer sequences (𝑠₀,𝑠₁,…,𝑠_𝑛−2) satisfying the constraints𝑐^−1∶𝑠₀+𝑠₁+⋯+𝑠_𝑛−3+𝑠_𝑛−2≤(𝑛−1)𝑛𝑐2𝑟−𝑙,_0∶2𝑠₀+𝑠₁+⋯+𝑠_𝑛−3+𝑠_𝑛−2≤(𝑛−1)𝑛𝑐2+1𝑟−𝑙,_1∶𝑠₀+𝑠₁+⋯+𝑠_𝑛−3+2+𝑠_𝑛−2≥(𝑛−1)𝑛𝑐2𝑘+1𝑟−𝑙,_𝑖∶𝑠_{𝑛−𝑖−1}−𝑠_𝑛−𝑖+𝑟≥0,𝑖∈ℤ/𝑛ℤ⧵{0,1}.(4.18) Hence, the dimension ℋ_𝑛𝑟 is just the number of nonnegative integer points contained in the polytope given by the system (4.18), where 𝑟=𝑠+1.

In dimension 3, one obtains the triangle in ℝ² given by the vertices 𝐴(0,2𝑟), 𝐵(𝑟,2𝑟),   and 𝐶(2𝑟,0) (see Section 3).

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

Figure 2 

An example of the polytope 𝑇4 in the case 𝑟=4. The vertices are in (−1/2,𝑟−1/2,2𝑟−1/2), (𝑟−2/3,2𝑟−2/3,3𝑟−2/3), (𝑟,2𝑟−1,3𝑟−1), (𝑟,2𝑟,3𝑟−2), (2𝑟−1/2,3𝑟−1/2,−1/2), and (3𝑟−1/2,−1/2,𝑟−1/2).

For the second method, one can observe that the dimension of ℋ_𝑛𝑟 is nothing else that the cardinality of the set 𝑆_𝐶 of all compositions (𝑠₀,…,𝑠_𝑛−1) of 𝑁=((𝑛−1)𝑛/2)𝑟−𝑙 subjected to the constraints (4.15). Therefore, if 𝑆_𝐶 is the set of all nonnegative integers (𝑠₀,…,𝑠_𝑛−1) satisfying the constraints (4.15) and 𝐹_𝐶 is the associated generating function, then the dimension of ℋ_𝑛𝑟 is the coefficient of 𝑞^𝑁 in 𝐹_𝐶(𝑞,𝑞,…,𝑞).The set 𝑆_𝐶 consists of all nonnegative integers points contained in the polytope of ℝ^𝑛𝒫_𝑛∶𝑥^{𝑛−𝑖−1}−𝑥_𝑛−𝑖+𝑟≥0,𝑖∈ℤ𝑥𝑛ℤ_𝑖≥0,𝑖∈ℤ𝑛ℤ.(4.19)

(See Figure 3).

Figure 3 

An example of the polytope 𝒫3 in the case 𝑟=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 “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.