Computing with Quantized Enveloping Algebras: PBW-Type Bases, Highest-Weight Modules andR-Matrices

Let Uq(g) be the quantized enveloping algebra corresponding to the semisimple Lie algebra g. We describe algorithms to obtain the multiplication table of a PBW-type basis of Uq(g). We use this to obtain an algorithm for calculating a Gröbner basis of an ideal in the subalgebraU − , which leads to a general construction of irreducible highest-weight modules over Uq(g). We also indicate how to compute the corresponding R -matrices.