On the Heisenberg invariance and the Elliptic Poisson tensors

TL;DR

This paper investigates the algebraic and geometric properties of Heisenberg invariant Poisson algebras, focusing on their unimodularity and classification, especially relating to elliptic Sklyanin-Odesskii-Feigin Poisson structures for dimensions up to six.

Contribution

It classifies all quadratic Heisenberg-invariant Poisson tensors on complex spaces of dimension up to six, revealing their connection to elliptic Sklyanin-Odesskii-Feigin algebras and their degenerations.

Findings

All such Poisson algebras are unimodular.

For dimensions up to five, they coincide with elliptic Sklyanin-Odesskii-Feigin algebras or their degenerations.

Complete classification of quadratic H-invariant Poisson tensors for n ≤ 6.

Abstract

We study different algebraic and geometric properties of Heisenberg invariant Poisson polynomial quadratic algebras. We show that these algebras are unimodular. The elliptic Sklyanin-Odesskii-Feigin Poisson algebras $q_{n,k}(\mathcal E)$ are the main important example. We classify all quadratic $H-$invariant Poisson tensors on ${\mathbb C}^n$ with $n\leq 6$ and show that for $n\leq 5$ they coincide with the elliptic Sklyanin-Odesskii-Feigin Poisson algebras or with their certain degenerations.