Matrix elements for indecomposable representations of complex su(2)

Abstract

Indecomposable representations of the simple complex Lie algebra A1 are investigated in this article from a general point of view. First a ‘‘master representation’’ is obtained which is defined on the space of the universal enveloping algebra Ω of A1. Then, from this master representation other indecomposable representations are derived which are induced on quotient spaces or subduced on invariant subspaces. Finally, it is shown that the familiar finite-dimensional and infinite-dimensional irreducible representations of su(2) and su(1,1) are closely related to certain of the indecomposable representations. Indecomposable representations of A1 [su(2), su(1,1)] have found increased applications in physical problems, including the unusual ‘‘finite multiplicity’’ indecomposable representations. Emphasis is placed in this article on an analysis of the more unfamiliar indecomposable representations. The matrix elements are obtained in explicit form for all representations which are discussed in this article. The methods used are purely algebraic.