On the Heisenberg invariance and the Elliptic Poisson tensors

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 $n\le 6$ and show that for $n\le 5$ they coincide with the elliptic Sklyanin-Odesskii-Feigin Poisson algebras or with their certain degenerations.

14 pages, no figures, minor revision, typos corrected

References in corpus (3)

• Calabi-Yau algebras
• Poisson (co)homology and isolated singularities
• The Van den Bergh duality and the modular symmetry of a Poisson variety

Cited by in corpus (3)

• Poisson modules and degeneracy loci
• Twisting Poisson algebras, coPoisson algebras and Quantization
• Quantised Painlevé monodromy manifolds, Sklyanin and Calabi-Yau algebras