జర్నల్ ఆఫ్ జెనరలైజ్డ్ లై థియరీ అండ్ అప్లికేషన్స్

On Contractions of Three-Dimensional Complex Associative Algebras

Abstract

Mohammed NF, Rakhimov IS and Sh Said Husain K

Contraction is one of the most important concepts that motivated by numerous applications in different fields of physics and mathematics. In this work, the contractions of complex associative algebras are considered. We focus on the variety A₃() of all complex associative algebras of dimension three (including nonunital). Various contractions criteria are collected and new criteria are proposed to test the possible existence of contraction for each pair of associative algebras. One of the main tools is the use of the low-dimensional cohomology groups of these algebras. As a result, we prove that the variety A₃() has seven irreducible components, two of dimension 5, four of dimension 7 and one of dimension 9.