2012/11/30 by Zhaoting Wei
Engineering · Mathematics · #Advanced Topics in Algebra #Algebra over a field #Algebraic structures and combinatorial models #Bracket #Engineering #Homotopy and Cohomology in Algebraic Topology #Infinitesimal #Lie algebra #Mathematical analysis #Mathematics #Noncommutative geometry #Order (exchange) #Physics #Poisson algebra #Poisson bracket #Poisson distribution #Poisson manifold #Pure mathematics #Quantization (signal processing) #Quantum #Quantum mechanics #Structural engineering #math.QA #math.RT #msc:16E40 #msc:17B63 #msc:22E46 #msc:53D55
paper · pdf · doi:10.1063/1.4927337
published as Journal of Mathematical Physics, 56 (2015), No.7, 071703 · 19 pages, minor changes of the previous version
openalex publication_date 2015/07/01 · arxiv created 2015/07/14 · openalex created_date 2016/06/24 · arxiv updated 2016/12/26 · openalex updated_date 2026/08/05
The family algebras are introduced by Kirillov in 2000. In this paper, we study the noncommutative Poisson bracket P on the classical family algebra 𝒞τ(𝔤). We show that P controls the first-order 1-parameter formal deformation from 𝒞τ(𝔤) to 𝒬τ(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.