Georgia Benkart • Rekha Biswal • Ellen Kirkman • Van C. Nguyen • Jieru Zhu

Tensor representations for the Drinfeld double of the Taft algebra

in Journal of 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 = –(q¹ˡ² + q⁻¹ˡ²) on the k-fold tensor power of any two-dimensional simple Dₙ-module. This action is known to be faithful for arbitrary k ≥ 1. We show that TLₖ(ξ) is isomorphic to the centralizer algebra End(Vᵏ) for 1 ≤ k ≤ 2n – 2.