Polynomial Poisson structures on affine solvmanifolds

A n-dimensional Lie group G equipped with a left invariant symplectic form ω+ is called a symplectic Lie group. It is well known that ω+ induces a left invariant affine structure on G. Relative to this affine structure we show that the left invariant Poisson tensor π+ corresponding to ω+ is polynomial of degree at most 1 and any right invariant k-multivector field on G is polynomial of degree at most k. If G is uni- modular, the symplectic form ω+ is also polynomial and the volume form ∧n/2ω+ 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.

• Autores: Medina Perea, Alirio Alberto; Boucetta, Mohamed
• Tipo de recurso: Article of investigation
• Fecha de publicación: 2011
• Institución: Universidad de Antioquia
• Repositorio: Repositorio UdeA
• Idioma: eng
• OAI Identifier: oai:bibliotecadigital.udea.edu.co:10495/45977
• Acceso en línea: https://hdl.handle.net/10495/45977
• Palabra clave: Lie groups; Symplectic geometry; Polimonios; Polynomials; Affine manifold; Polynomial tensors; http://id.loc.gov/authorities/subjects/sh85076786; http://id.loc.gov/authorities/subjects/sh2002004420; http://id.loc.gov/authorities/subjects/sh85104702
• Rights: openAccess
• License: http://creativecommons.org/licenses/by-nc-sa/4.0/