A counterexample to the existence of a Poisson structure on a twisted group algebra

Abstract

Crawley-Boevey [1] introduced the definition of a noncommutative Poisson structure on an associative algebra

A that extends the notion of the usual Poisson bracket. Let (V, w) be a symplectic mani-fold and G be a finite group of symplectimorphisms of V. Consider the twisted group algebra A = C[V ]#G. We produce a counterexample to prove that it is not always possible to define a noncommutative poisson structure on C[V ]#G that extends the Poisson bracket on C[V ]G.