Factorization of Simple Modules for Certain Pointed Hopf Algebras

We study the representations of two types of pointed Hopf algebras: restricted two-parameter quantum groups, and the Drinfel’d doubles of rank one pointed Hopf algebras of nilpotent type. We study, in particular, under what conditions a simple module can be factored as the tensor product of a one dimensional module with a module that is naturally a module for the quotient by central group-like elements. For restricted two-parameter quantum groups, given θ a primitive "th root of unity, the factorization of simple uθy,θz (sln)-modules is possible, if and only if gcd((y − z)n, ") = 1. For rank one pointed Hopf algebras, given the data D = (G,χ, a), the factorization of simple D(HD)-modules is possible if and only if |χ(a)| is odd and |χ(a)| = |a| = |χ|. Under this condition, the tensor product of two simple D(HD)-modules is completely reducible, if and only if the sum of their dimensions is less than or equal to |χ(a)| + 1.