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 Gröbner 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 image4 are discussed.