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 Drinfelʹd 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 ℓth root of unity, the factorization of simple -modules is possible, if and only if gcd((y−z)n,ℓ)=1. For rank one pointed Hopf algebras, given the data , the factorization of simple -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 -modules is completely reducible, if and only if the sum of their dimensions is less than or equal to χ(a)+1.