Generic Modules Over a Class of Drinfeld's Quantum Doubles

Abstract

Let k be a field and A n be the Taft's n 2 -dimensional Hopf algebras. When n is odd, the Drinfeld quantum double D A n of A n is a Ribbon Hopf algebra. In the previous articles, we constructed an n 4 -dimensional Hopf algebra H n p q which is isomorphic to D A n if p = 0 and q = −1 , and studied the finite dimensional representations of H n 1 q . We showed that the basic algebra of any nonsimple block of H n 1 q is independent of n. In this article, we examine the infinite representations of H 2 1 −1 , or equivalently of H n 1 q D A n for any n ≥ 2. We investigate the indecomposable and algebraically compact modules over H 2 1 −1 , describe the structures of these modules and classify them under the elementary equivalence.