Constructing representations of split semisimple Lie algebras

We describe an algorithm for constructing irreducible representations of split semisimple Lie algebras in characteristic 0. The algorithm calculates a Grobner basis of a certain left ideal in a universal enveloping algebra, It is shown that this algorithm runs in polynomial time if the Lie algebra is fixed. At the end of the paper, practical experiences with an implementation of the algorithm in GAP4 are discussed. (C) 2001 Elsevier Science B.V. All rights reserved.