Commutative Poisson algebras from deformations of noncommutative algebras
type: Item
instance of: scholarly article
title: Commutative Poisson algebras from deformations of noncommutative algebras
author: Alexander V. Mikhailov
author: Pol Vanhaecke
published in: Letters in Mathematical Physics
publication date: 2024-10-22T00:00:00Z
review text: It is well known that ``taking the classical limit'', the counterpart of deformation quantization, may fail for a noncommutative phase space \(\mathcal{A}\). The difficulty in the noncommutative case is that the limit of commutators of operators provides a Hamiltonian structure which may be incompatible with the multiplication in \(\mathcal{A}\), making it impossible to obtain a Poisson algebra. \N\NThe authors introduce a generalization of the Hamiltonian structure, which deals with the above difficutly. Specifically, they view \(\mathcal{A}\) as a Poisson module over a commutative Poisson algebra structure on \(\Pi(\mathcal{A}) = Z(\mathcal{A}) \times (\mathcal{A} / Z\mathcal{A})\), which arises naturally from the deformation.
reviewed by: Iakovos Androulidakis
Mathematics Subject Classification ID: 53D55
Mathematics Subject Classification ID: 53D17
Mathematics Subject Classification ID: 17B63
Mathematics Subject Classification ID: 17B80
Mathematics Subject Classification ID: 37J37
zbMATH DE Number: 7930028
zbMATH Keywords: Poisson algebras
zbMATH Keywords: Poisson modules
zbMATH Keywords: deformations of noncommutative algebras
zbMATH Keywords: deformation quantisation
zbMATH Keywords: Heisenberg derivations
zbMATH Keywords: Hamiltonian derivations
MaRDI profile type: Publication
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DOI: 10.1007/S11005-024-01855-3
description: scientific article; zbMATH DE number 7930028