# 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 $n\leq 6$ 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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https://hal.archives-ouvertes.fr/hal-00619673
Submitted on : Tuesday, September 6, 2011 - 5:05:29 PM
Last modification on : Monday, March 9, 2020 - 6:15:53 PM

### Identifiers

• HAL Id : hal-00619673, version 1
• ARXIV : 1001.4422

### Citation

G. Ortenzi, V. Rubtsov, S. R. Tagne Pelap. On the Heisenberg invariance and the Elliptic Poisson tensors. 2010. ⟨hal-00619673⟩

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