Tensor Representations for the Drinfeld Double of the Taft Algebra

For the Drinfeld double $D_n$ of the Taft algebra $A_n$ defined over an algebraically closed field $mathbb k$ of characteristic zero using a primitive $n$th root of unity $q in mathbb k$ for $n$ odd, $nge3$, we determine the ribbon element of $D_n$ explicitly. We use the R-matrix and ribbon element of $D_n$ to construct an action of the Temperley-Lieb algebra $mathsf{TL}_k(xi)$ with $xi = -(q^{frac{1}{2}}+q^{-frac{1}{2}})$ on the $k$-fold tensor product $V^{otimes k}$ of any two-dimensional simple $D_n$-module $V$. When $V$ is the unique self-dual two-dimensional simple module, we develop a diagrammatic algorithm for computing the $mathsf{TL}_k(xi)$-action. We show that this action on $V^{otimes k}$ is faithful for arbitrary $k ge 1$ and that $mathsf{TL}_k(xi)$ is isomorphic to the centralizer algebra $text{End}_{D_n}(V^{otimes k})$ for $1 le kle 2n-2$.