  Let $U$ and $A$ be algebras over a field $k$. We study algebra structures $H$
on the underlying tensor product $U{\otimes}A$ of vector spaces which satisfy
$(u{\otimes}a)(u'{\otimes}a') = uu'{\otimes}aa'$ if $a = 1$ or $u' = 1$. For a
pair of characters $\rho \in \Alg(U, k)$ and $\chi \in \Alg(A, k)$ we define a
left $H$-module $L(\rho, \chi)$. Under reasonable hypotheses the correspondence
$(\rho, \chi) \mapsto L(\rho, \chi)$ determines a bijection between character
pairs and the isomorphism classes of objects in a certain category
${}_H\underline{\mathcal M}$ of left $H$-modules. In many cases the
finite-dimensional objects of ${}_H\underline{\mathcal M}$ 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.
