Deformation quantization of Poisson algebras
type: Item
instance of: scholarly article
title: Deformation quantization of Poisson algebras
zbMATH Open document ID: 0768.58018
author: Hideki Omori
author: Yoshiaki Maeda
author: Akira Yoshioka
published in: Proceedings of the Japan Academy. Series A
publication date: 1993-01-17T00:00:00Z
review text: \textit{A. Lichnerowicz} [J. Differ. Geom. 12, 253-300 (1977; Zbl 0405.53024)] (and others) proposed the concept of a deformation quantization of a symplectic manifold as a formal deformation of the algebra \({\mathfrak a} = C^ \infty(M)\). The authors [Adv. Math. 85, 224-255 (1991; Zbl 0734.58011)] proved that any symplectic manifold is deformation quantizable. In the paper under review, the authors define analogously the notion of a deformation quantization of order \(k\) (cutting the space of formal series to \({\mathfrak a} \oplus \nu{\mathfrak a}\oplus\dots\oplus \nu^ k{\mathfrak a}\) for an arbitrary Poisson structure \(\{\cdot,\cdot\}\) on a manifold \(M\)). Next, they study the obstructions for a deformation quantization of order \(k\) to be extended to that of order \(k+1\). The obstruction is obtained as a de Rham-Chevalley 3- cocycle.
Mathematics Subject Classification ID: 37J99
Mathematics Subject Classification ID: 17B56
Mathematics Subject Classification ID: 17B65
Mathematics Subject Classification ID: 53D50
zbMATH DE Number: 97938
zbMATH Keywords: Poisson bracket
zbMATH Keywords: Hochschild coboundary operator
zbMATH Keywords: de Rham-Chevalley cohomology group
zbMATH Keywords: deformation quantization
zbMATH Keywords: obstructions
reviewed by: Jan Kubarski
MaRDI profile type: Publication
full work available at URL: pjaa.68.97
OpenAlex ID: W1980062029
cites work: Existence of star-products and of formal deformations of the Poisson Lie algebra of arbitrary symplectic manifolds
cites work: Les variétés de Poisson et leurs algèbres de Lie associees
cites work: Q5343374
cites work: Weyl manifolds and deformation quantization
cites work: C*-algebras associated with irrational rotations
cites work: Poisson manifolds and the Schouten bracket
DOI: 10.3792/PJAA.68.97
description: scientific article; zbMATH DE number 97938