Computing Representations of a Lie Group via the Universal Enveloping Algebra

We present an algorithm for the computation of representations of a Lie algebra acting on its universal enveloping algebra. This is a new algorithm which permits the effective computation of these representations and of the matrix elements of the corresponding Lie group. The approach is based on a mathematical formulation originated by the authors. An interesting feature is the efficient computation of the adjoint representation of the corresponding Lie group. The methods are implemented using a symbolic computation program such as MAPLE.