The noncommutative Poisson bracket and the deformation of the family algebras

TL;DR

This paper investigates the noncommutative Poisson bracket on classical family algebras, demonstrating it as a first-order deformation to quantum algebras, and explores related quantization issues.

Contribution

It introduces the noncommutative Poisson bracket as a deformation of family algebras and proves its triviality as a Hochschild 2-coboundary.

Findings

The noncommutative Poisson bracket is a Hochschild 2-coboundary.

The deformation from classical to quantum family algebra is infinitesimally trivial.

Discussion on Mackey's analogue and quantization challenges.

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.