Generic Modules Over a Class of Drinfeld's Quantum Doubles

Abstract

Let k be a field and A n (ω) be the Taft's n 2-dimensional Hopf algebras. When n is odd, the Drinfeld quantum double D(A n (ω)) of A n (ω) 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 (ω)) if p ≠ 0 and q = ω−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 (ω)) 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.