A comparison of deformations and geometric study of associative algebras varieties

Abstract

The aim of this paper is to give an overview and to compare the different deformation theories of algebraic structures. We describe in each case the corresponding notions of degeneration and rigidity. We illustrate these notions with examples and give some general properties. The last part of this work shows how these notions help in the study of associative algebras varieties. 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 of 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 has in common the fact that we make an "extension" of the field. These theories may be applied to any multilinear structure. We shall be concerned in this paper with the category of associative algebras.