Deformation quantization of polynomial Poisson algebras
type: Item
instance of: scholarly article
title: Deformation quantization of polynomial Poisson algebras
zbMATH Open document ID: 0986.17007
author: Michael Penkava
author: Pol Vanhaecke
published in: Journal of Algebra
publication date: 2001-10-29T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/9804022
review text: An explicit third order deformation quantization of an arbitrary polynomial Poisson algebra is provided. The second and third order terms are obtained formally in the same way as analogous terms of the star-product induced by the universal enveloping algebra in the linear case. For instance, the second term for the Poisson bracket \(\pi_1=\frac{1}{2}X_{ij} \partial^i\wedge\partial^j\) reads \[ \pi_2=\frac{1}{6}X^l_{ij}X_{lk}\partial^i\cdot\partial^{jk}+ \frac{1}{8}X_{ij}X_{kl}\partial^{ik}\cdot\partial^{jl}, \] where \(\cdot\) is the tensor symmetrization. This deformation quantization differs from the Kontsevich one and not always extends to the order four. A correction term is computed which cancels this obstruction.
Mathematics Subject Classification ID: 17B63
Mathematics Subject Classification ID: 53D55
zbMATH DE Number: 1448648
zbMATH Keywords: deformation quantization
zbMATH Keywords: Poisson algebra
zbMATH Keywords: universal enveloping algebra
reviewed by: J. Grabowski
MaRDI profile type: Publication
OpenAlex ID: W2056816526
arXiv ID: math/9804022
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DOI: 10.1006/JABR.1999.8239
description: scientific article