Tensor representations for the Drinfeld double of the Taft algebra

Abstract

Over an algebraically closed field k of characteristic zero, the Drinfeld double Dn of the Taft algebra that is defined using a primitive nth root of unity q∈k for n≥2 is a quasitriangular Hopf algebra. Kauffman and Radford have shown that Dn has a ribbon element if and only if n is odd, and the ribbon element is unique; however there has been no explicit description of this element. In this work, we determine the ribbon element of Dn explicitly. For any n≥2, we use the R-matrix of Dn to construct an action of the Temperley-Lieb algebra TLk(ξ) with ξ=−(q12+q−12) on the k-fold tensor power V⊗k of any two-dimensional simple Dn-module V. This action is known to be faithful for arbitrary k≥1. We show that TLk(ξ) is isomorphic to the centralizer algebra EndDn(V⊗k) for 1≤k≤2n−2.