Polynomial Poisson structures on affine solvmanifolds

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dc.creatorMedina, Mohamed Boucetta-Alberto
dc.date2008-02-04
dc.date.accessioned2026-07-07T09:18:32Z
dc.date.available2026-07-07T09:18:32Z
dc.descriptionA $n$-dimensional Lie group $G$ equipped with a left invariant symplectic form $\om^+$ is called a symplectic Lie group. It is well-known that $\om^+$ induces a left invariant affine structure on $G$. Relatively to this affine structure we show that the left invariant Poisson tensor $π^+$ corresponding to $\om^+$ is polynomial of degree 1 and any right invariant $k$-multivector field on $G$ is polynomial of degree at most $k$. If $G$ is unimodular, the symplectic form $\om^+$ is also polynomial and the volume form $\wedge^{\frac{n}2}\om^+$ is parallel. We show also that any left invariant tensor field on a nilpotent symplectic Lie group is polynomial, in particular, any left invariant Poisson structure on a nilpotent symplectic Lie group is polynomial. Because many symplectic Lie groups admit uniform lattices, we get a large class of polynomial Poisson structures on compact affine solvmanifolds.
dc.description17 pages
dc.identifierhttps://arxiv.org/abs/0802.0357
dc.identifierhttp://arxiv.org/abs/0802.0357
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/154061
dc.subjectSymplectic Geometry
dc.subjectDifferential Geometry
dc.subject53D05, 53D17
dc.titlePolynomial Poisson structures on affine solvmanifolds
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