Representations Parameterized by a Pair of Characters

Abstract

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 $(u\otimes a)(u^{\prime }\otimes a^{\prime })=u{u}^{\prime }\otimes a{a}^{\prime }$ if $a=1$ or $u^{\prime }=1$. For a pair of characters $ρ\in \Alg(U, k)$ and $χ\in \Alg(A, k)$ 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.

amstex, 33 pages

References in corpus (1)

• On the irreducible representations of a class of pointed Hopf algebras

Cited by in corpus (1)

• A class of quantum doubles which are ribbon algebras