skip to main content

Title: The noncommutative Poisson bracket and the deformation of the family algebras

The family algebras are introduced by Kirillov in 2000. In this paper, we study the noncommutative Poisson bracket P on the classical family algebra C{sub τ}(g). We show that P controls the first-order 1-parameter formal deformation from C{sub τ}(g) to Q{sub τ}(g) where the latter is the quantum family algebra. Moreover, we will prove that the noncommutative Poisson bracket is in fact a Hochschild 2-coboundary, and therefore, the deformation is infinitesimally trivial. In the last part of this paper, we discuss the relation between Mackey’s analogue and the quantization problem of the family algebras.
Authors:
 [1]
  1. Department of Mathematics, Indiana University, 831 E 3rd Street, Bloomington, Indiana 47405 (United States)
Publication Date:
OSTI Identifier:
22479575
Resource Type:
Journal Article
Resource Relation:
Journal Name: Journal of Mathematical Physics; Journal Volume: 56; Journal Issue: 7; Other Information: (c) 2015 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA)
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ALGEBRA; COMMUTATION RELATIONS; QUANTIZATION; QUANTUM GROUPS