Representations Parameterized by a Pair of Characters

Let U and A be algebras over a field k. We study algebra structures H on the underlying tensor product U⊗A of vector spaces which satisfy (u⊗a)(u′⊗a′) = uu′⊗aa′ if a = 1 or u′ = 1. For a pair of characters ρ ∈ Alg(U, k) and χ ∈ Alg(A, k) we define a left H-module L(ρ, χ). Under reasonable hypotheses the correspondence (ρ, χ) 7→ L(ρ, χ) determines a bijection between character pairs and the isomorphism classes of objects in a certain category HM of left H-modules. In many cases the finite-dimensional objects of HM are the finite-dimensional irreducible left H-modules. We show that the finite-dimensional irreducible representations of a wide class of pointed Hopf algebras are parameterized by pairs of characters.