Abstract
The aim of this paper is to give an overview and to compare the different deformation theories of algebraic structures. In each case we describe the corresponding notions of degeneration and rigidity. We illustrate these notions by examples and give some general properties. The last part of this work shows how these notions help in the study of varieties of associative algebras. The first and popular deformation approach was introduced by M. Gerstenhaber for rings and algebras using formal power series. A noncommutative version was given by Pinczon and generalized by F. Nadaud. A more general approach called global deformation follows from a general theory by Schlessinger and was developed by A. Fialowski in order to deform infinite-dimensional nilpotent Lie algebras. In a nonstandard framework, M. Goze introduced the notion of perturbation for studying the rigidity of finite-dimensional complex Lie algebras. All these approaches share the common fact that we make an “extension” of the field. These theories may be applied to any multilinear structure. In this paper, we will be dealing with the category of associative algebras.
Highlights
Throughout this paper K will be an algebraically closed field, and Ꮽ denotes an associative K-algebra
In [9], we show the existence of associative deformation of a universal enveloping algebra using the linear Poisson structure of the Lie algebra
Goze et al introduced the notion of perturbation for studying the rigidity of Lie algebras [14, 15], it was used to describe the 6-dimensional rigid associative algebras [16, 17]
Summary
Throughout this paper K will be an algebraically closed field, and Ꮽ denotes an associative K-algebra. A noncommutative version, where the parameter no longer commutes with the element of algebra, was introduced by Pinczon [12] and generalized by Nadaud [13] They describe the corresponding cohomology and show that the Weyl algebra, which is rigid for formal deformation, is nonrigid in the noncommutative case. Goze et al introduced the notion of perturbation for studying the rigidity of Lie algebras [14, 15], it was used to describe the 6-dimensional rigid associative algebras [16, 17]. A more general notion called global deformation was introduced by Fialowski, following Schlessinger, for Lie algebras [18]. The perturbation approach is sometimes more adapted to direct computation
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