Finite-Dimensional Representations of a Quantum Double

Abstract

Let k be a field and let An(ω) be the Taft's n2-dimensional Hopf algebra. When n is odd, the Drinfeld quantum double D(An(ω)) of An(ω) is a ribbon Hopf algebra. In a previous paper we constructed an n4-dimensional Hopf algebra Hn(p, q) which is isomorphic to D(An(ω)) if p ≠ 0 and q = ω− 1, and studied the irreducible representations of Hn(1, q). We continue our study of Hn(p, q), and we examine the finite-dimensional representations of H3(1, q), equivalently, of D(A3(ω)). We investigate the indecomposable left H3(1, q)-module, and describe the structures and properties of all indecomposable modules and classify them when k is algebraically closed. We also give all almost split sequences in mod H3(1, q), and the Auslander–Reiten quiver of H3(1, q).