Multiplicity free and finite multiplicity indecomposable representations of the algebra su(1,1)

Abstract

A classification is given for the multiplicity free indecomposable representations of the simple Lie algebra su(1,1), which are unbounded on both sides. Formulas have been obtained for the matrix elements of the generators of su(1,1) for all these representations. Representations of su(1,1) are analyzed which have the property that all their weight subspaces are infinite dimensional. Subrepresentations and representations on quotient spaces of this infinite multiplicity representations are considered and their relationship to the multiplicity free indecomposable representations is determined (both, unbounded on both sides, and bounded on one side). Finite multiplicity indecomposable representations are obtained from the infinite multiplicity representation for special values of the Casimir operator. A decomposition of the infinite multiplicity representation into a direct sum of multiplicity free representations and finite multiplicity indecomposable respresentations is given in two different ways. Finally, formulas for the matrix elements of su(1,1) are given for the finite multiplicity indecomposable representations.