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

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(mod ℤ_n) and by the operators x → ɛⁱx_i, where ɛⁿ = 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 ℙⁿ. 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].

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.

2.3. 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:

()

2.3.2. 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 ɛ = e^2πi/n.

Consider σ, τ of GL(V) defined by

$$ ()

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 = ℂⁿ, 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:

$$ ()

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 $$ 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:

$$ ()

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},

$$ ()

3. H-Invariant JPS in Dimension 3

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.

Set $$, where $$. 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.

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.

4.2. 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

$$ ()

for f, g ∈ ℛ.

We will denote by P the particular Casimir $$ 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

$$ ()

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

$$ ()

From the τ-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

$$ ()

α(i₁) + 1 ≠ ⋯≠α(i_n−2) + 1 ∈ Z/nℤ∖{i + 1, j + 1} and the τ-degree condition.

Set

$$ ()

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,

$$ ()

Hence,

$$ ()

Since τ − ϖ(σ · Q) = τ − ϖ(Q) = l,  i ≡ 0  modulo  n.

Set $$. 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

$$ ()

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

$$ ()

where r = s + 1 and the s₀, …, s_n−1 satisfy the condition

$$ ()

Therefore α₀, α₁, …, α_n−1 are completely determined by the set of nonnegative integer sequences (s₀, s₁, …, s_n−1) satisfying the constraints

$$ ()

and such that

$$ ()

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

$$ ()

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 constraints

$$ ()

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).

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 ℝⁿ

$$ ()

(See Figure 3).