International Journal of Mathematics and Mathematical Sciences
Volume 2007 (2007), Article ID 18915, 24 pages
Research Article

A Comparison of Deformations and Geometric Study of Varieties of Associative Algebras

Abdenacer Makhlouf

Laboratoire de Mathématiques, Informatique et Application, Université de Haute Alsace, 4 rue des Frères Lumière, Mulhouse Cedex 68093, France

Received 13 May 2005; Accepted 7 February 2007

Academic Editor: Howard E. Bell

Copyright © 2007 Abdenacer Makhlouf. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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.