Representations of the Euclidean group in the plane

Abstract

The theory of representations of the Euclidean algebra and group in the planeE2 is investigated from a very general and basic point of view. A master representation is obtained for both the algebra as well as for the group. This master representation induces (subduces) all the representations ofE2 which are studied in this paper. The master representation is defined on the space of the universal enveloping algebra Ω ofE2, whose basis vectors are labelled by three (discrete) parameters. The finite-dimensional (linear, indecomposable) representations which are induced by the master representation are classified, both for the algebra and for the group. The bases for these representations are given and for the group the finite matrix elements. While the result regarding the classification of thefinite-dimensional indecomposable representations has been published, without proofs, we give here the proofs of the theorems. Moreover, the three- and four-dimensional representations of the Euclidean algebra and group, which are induced by the master representation, are obtained in this paper in explicit form.Infinite-dimensional indecomposable representations which are induced by the master representation are discussed at the level of both the algebra and the group. The familiar result of the infinite-dimensional irreducible representations is obtained as a final, and rather special, case of representations induced by the (indecomposable) master representation. The matrix elements for all representations of the groupE2 which are discussed here are given in explicit form. Due to the particular approach taken in this paper, relations among functions representing matrix elements in different representations are obtained.