Universal deformation formulas and breaking symmetry
type: Item
instance of: scholarly article
title: Universal deformation formulas and breaking symmetry
zbMATH Open document ID: 0827.17021
DOI: 10.1016/0022-4049(93)90041-Q
author: Murray Gerstenhaber
author: Samuel D. Schack
author: Vincent Coll
published in: Journal of Pure and Applied Algebra
publication date: 1995-12-17T00:00:00Z
review text: Let \(A\) be an algebra over an algebraically closed field of characteristic zero (not necessarily associative or Lie). Let \(\varphi\) and \(\psi\) be two commuting derivations of \(A\). The authors define a new multiplication on the algebra \(At\) of formal power series on \(A\) by using the following formula: \[ a*_t b= ab+ t\varphi (a) \psi(b)+ {\textstyle {t^2 \over {2!}}} \varphi^2 (a) \psi^2 (b)+ \cdots\;. \] If \(A\) is associative this gives a one-parameter family of associative multiplications. The authors then observe that if \(A\) has a Lie group of automorphisms of dimension at least two, then \(A\) may be deformed by using the above formula in such a way that the full group no longer acts on \(A\). They call this phenomenon ``breaking symmetry''.
Mathematics Subject Classification ID: 17B56
zbMATH DE Number: 500583
zbMATH Keywords: universal deformation
zbMATH Keywords: breaking symmetry
zbMATH Keywords: Lie group of automorphisms
reviewed by: Alexander Premet
MaRDI profile type: Publication
cites work: Filtered and associated graded rings
cites work: Q4733981
cites work: Q4735825
cites work: Q5749433
cites work: The cohomology structure of an associative ring
cites work: On the deformation of rings and algebras
cites work: On the deformation of rings and algebras. III
cites work: Quantum symmetry
cites work: Relative Hochschild cohomology, rigid algebras, and the Bockstein
cites work: Q3831155
cites work: Bialgebra cohomology, deformations, and quantum groups.
cites work: Q4693937
cites work: Abstract Jacobi and Poisson structures. Quantization and star-products
cites work: Differential Forms On Regular Affine Algebras
cites work: Q3313005
cites work: Q5788173
cites work: Déformation du crochet de Poisson sur une variété symplectique
Recommended article: Universal Deformation Formulas
Recommended article: Universal deformation formulae, symplectic Lie groups and symmetric spaces
Recommended article: Universal deformation formula, formality and actions
Recommended article: Symmetry breaking by deformations
Recommended article: Dihedral universal deformations
Recommended article: Unifying derived deformation theories
Recommended article: Symmetries in connection preserving deformations
Recommended article: Deformation Theory
Recommended article: Deformation from symmetry and multiplicity of solutions in non-homogeneous problems
Recommended article: Deformations in mathematics and physics
description: scientific article; zbMATH DE number 500583