Computing Representations of a Lie Group via the Universal Enveloping Algebra

The technique of computing representations on the enveloping algebra was developed in work by physicists, mainly Gruber (e.g. Gruber and Klimyk (1986)). Their methods are not designed for computations with Lie algebras beyond the simplest ones. Our idea is to construct left and right principal matrices, corresponding to left and right multiplication of the basis elements of the Lie algebra on the basis of monomials of the universal enveloping algebra (see, e.g. Dixmier (1977), Bourbaki (1971–72)). These allow one to compute all representations of the Lie algebra, and the matrix elements of the corresponding Lie group, acting on the universal enveloping algebra and its quotients. We show how to compute these representations, which are typically infinite-dimensional. As the universal enveloping algebra is non-commutative, it is not an easy problem to calculate products in that algebra. Our method resolves these difficulties, by using the matrices defining the Lie algebra directly. In the next section we present some notations and the theoretical basis of the algorithm. Then we present our algorithm along with the example of E2 (Euclidean group in two dimensions) which comes up naturally in many physical problems. The calculation of the adjoint group (see Chevalley (1946)) comes for free in our approach. Other applications include calculating the matrix elements of the representations as well as recurrence relations for the matrix elements. We include some remarks concerning the efficiency of our approach.