Tensor Representations for the Drinfeld Double of the Taft Algebra

Over an algebraically closed field $\mathbb{k}$ of characteristic zero, the Drinfeld double $D_n$ of the Taft algebra that is defined using a primitive $n$th root of unity $q\in \mathbb{k}$ for $n\ge 2$ is a quasitriangular Hopf algebra. Kauffman and Radford have shown that $D_n$ 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 $D_n$ explicitly. For any $n\ge 2$, we use the R-matrix of $D_n$ to construct an action of the Temperley-Lieb algebra ${\mathsf{T}\mathsf{L}}_k(\xi )$ with $\xi =-(q^{\frac{1}{2}}+q^{-\frac{1}{2}})$ on the $k$-fold tensor power $V^{\otimes k}$ of any two-dimensional simple $D_n$-module $V$. This action is known to be faithful for arbitrary $k\ge 1$. We show that ${\mathsf{T}\mathsf{L}}_k(\xi )$ is isomorphic to the centralizer algebra ${\text{End}}_{D_n}(V^{\otimes k})$ for $1\le k\le 2n-2$.