The noncommutative Poisson bracket and the deformation of the family algebras

Journal Article · Wed Jul 15 00:00:00 EDT 2015 · Journal of Mathematical Physics

DOI:https://doi.org/10.1063/1.4927337· OSTI ID:22479575

Wei, Zhaoting ^[1]

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1. Department of Mathematics, Indiana University, 831 E 3rd Street, Bloomington, Indiana 47405 (United States)

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.

• OSTI ID: 22479575
• Journal Information: Journal of Mathematical Physics, Vol. 56, Issue 7; Other Information: (c) 2015 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA); ISSN 0022-2488
• Country of Publication: United States
• Language: English

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
ALGEBRA
COMMUTATION RELATIONS
QUANTIZATION
QUANTUM GROUPS