Q-Deformation of Semisimple and Non-Semisimple Lie Algebras

Abstract

We give an elementary introduction to the Drinfeld-Jimbo procedure of the quantum deformation U q (g) of semisimple Lie algebras g. The q—Serre relations are discussed in some detail, in the Chevalley and the Cartan—Weyl basis. We review the real forms of g and U q (g). The general procedure is illustrated by the examples U q (sl(2)) and U q (so(5, ℂ)). After some general considerations on non-semisimple Lie algebras, we discuss in detail the quantum deformation of the Poincare algebra, involving the contraction of U q (so(3, 2)). Some physical consequences are mentioned.