Unitary highest weight representation of Uq(su(1,1)) when q is a root of unity

Abstract

Unitary representation of Uq(su(1,1)) with q being roots of unity are studied. The authors construct unitary irreducible highest weight modules and find that the representations are discrete series owing to the unitarity. Moreover, it is revealed that each unitary irreducible highest weight module is equivalent to the tensor product of two modules. They show that one of them is just the unitary irreducible highest weight module of su(1,1) and the other is the same as that of Uq(su(2)). It is also shown that the number of the latter modules is finite in their representation.