Tensor Representations for the Drinfeld Double of the Taft Algebra

TL;DR

This paper explicitly determines the ribbon element of the Drinfeld double of the Taft algebra and explores its tensor representations, connecting the Temperley-Lieb algebra with the centralizer algebra of tensor powers.

Contribution

It provides an explicit description of the ribbon element for the Drinfeld double of the Taft algebra and establishes an isomorphism between the Temperley-Lieb algebra and the centralizer algebra of tensor powers.

Findings

Explicit ribbon element for D_n determined

Faithful action of Temperley-Lieb algebra on tensor powers

Isomorphism between TL_k and centralizer algebra for 1 ≤ k ≤ 2n-2

Abstract

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 $n \geq 2$ 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 $n \geq 2$, 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 is known 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 $1 \le k \le 2n-2$.