On the Heisenberg invariance and the Elliptic Poisson tensors

Details On the Heisenberg invariance and the Elliptic Poisson tensors On the Heisenberg invariance and the Elliptic Poisson tensors

2010-01-25

arxiv.org/abs/1001.4422v2

Physics

Mathematical Physics

14 pages, no figures, minor revision, typos corrected

Scientific paper

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.

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