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.

19 pages, minor changes of the previous version

References in corpus (3)

• Formality theorems for Hochschild complexes and their applications
• Covariant Weil algebras
• The Classical Family Algebra of the Adjoint Representation of $sl(n)$

Cited by in corpus (1)

• Commutative avatars of representations of semisimple Lie groups