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Over an algebraically closed field $\mathbb k$ of characteristic zero, theDrinfeld double $D_n$ of the Taft algebra that is defined using a primitive$n$th root of unity $q \in \mathbb k$ for $n \geq 2$ is a quasitriangular Hopfalgebra. Kauffman and Radford have shown that $D_n$ has a ribbon element if andonly if $n$ is odd, and the ribbon element is unique; however there has been noexplicit description of this element. In this work, we determine the ribbonelement of $D_n$ explicitly. For any $n \geq 2$, we use the R-matrix 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 power$V^{\otimes k}$ of any two-dimensional simple $D_n$-module $V$. This action isknown to be faithful for arbitrary $k \geq 1$. We show that$\mathsf{TL}_k(\xi)$ is isomorphic to the centralizer algebra$\text{End}_{D_n}(V^{\otimes k})$ for $1 \le k \le 2n-2$.

Tensor Representations for the Drinfeld Double of the Taft Algebra