Tensor products of Poisson algebras
type: Item
instance of: scholarly article
title: Tensor products of Poisson algebras
zbMATH Open document ID: 0827.17022
DOI: 10.1007/BF01209563
author: V. P. Cherkashin
published in: Mathematical Notes
publication date: 1994-11-16T00:00:00Z
review text: The author considers cohomologies, deformations, and homotopies of tensor products of Poisson algebras with applications to homogeneous (in particular quadratic) brackets. Let \((P_i, \mu_i, [ ,\;]_i)\), \(i=1,2\) be Poisson algebras, \(M_i\) modules over \(P_i\), and \(D^\bullet (P_i,M_i)\) the associated complexes. The author constructs a morphism of complexes \[ \iota: D^\bullet (P_1, M_1) \otimes D^\bullet (P_2, M_2)\to D^\bullet (P_1 \otimes P_2, M_1 \otimes M_2). \] He then associates to each bi-derivation \(\theta: P_1\otimes P_2\to P_1 \otimes P_2\) (subject to a symmetry condition) a new Poisson algebra \((P_1 \otimes P_2, \mu_\otimes, [ ,\;]_\theta)\) characterized by the requirements that \(P_1 \otimes 1\), \(1\otimes P_2\) be subalgebras and \([x \otimes 1, 1\otimes y]_\theta= \theta (x\otimes y)\). As an application, exact Poisson deformations of \(P_1\otimes P_2\) are constructed. Formulae for homotopies of \(P_1\otimes P_2\) by means of central elements (with applications to \(P_1\) homogeneous and \(P_2\) trivial) are given. The author then combines homotopies and the above \(\theta\)- construction to construct yet other classes of Poisson brackets.
Mathematics Subject Classification ID: 17B56
Mathematics Subject Classification ID: 37J99
Mathematics Subject Classification ID: 17A45
zbMATH DE Number: 687512
reviewed by: Serge Parmentier
MaRDI profile type: Publication
cites work: Q5335321
cites work: La structure de Poisson sur l'algèbre symétrique d'une algèbre de Lie nilpotente
cites work: Q4002726
cites work: Q5680794
cites work: Q3760711
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full work available at URL: https://doi.org/10.1007/bf01209563
OpenAlex ID: W1998394106
description: scientific article; zbMATH DE number 687512