The noncommutative Poisson bracket and the deformation of the family algebras

Abstract

A.A. Kirillov introduced the family algebras in 2000. In this paper we study the noncommutative Poisson bracket P on the classical family algebra. We show that P is the first-order deformation from the classical family algebra to the quantum family algebra. We will prove that the noncommutative Poisson bracket is in fact a Hochschild 2-coboundary therefore the deformation is infinitesimally trivial. In the last part of this paper we also talk about Mackey's analogue and the quantization problem of the family algebras.