Representations Parameterized by a Pair of Characters

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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∈ (U, k) and χ∈ (A, k) we define a left H-module L(ρ, χ). Under reasonable hypotheses the correspondence (ρ, χ) 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. In math.QA/0603269 we apply the results of this paper and show that the finite-dimensional irreducible representations of a wide class of pointed Hopf algebras are parameterized by pairs of characters.