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#### Tensor representations for the Drinfeld double of the Taft algebra

##### External Resource

https://doi.org/10.1016/j.jalgebra.2022.04.041

(Publisher version)

https://doi.org/10.48550/arXiv.2012.15277

(Preprint)

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##### Citation

Benkart, G., Biswal, R., Kirkman, E., Nguyen, V. C., & Zhu, J. (2022). Tensor representations
for the Drinfeld double of the Taft algebra.* Journal of Algebra,* (606), 764-797. doi:10.1016/j.jalgebra.2022.04.041.

Cite as: https://hdl.handle.net/21.11116/0000-000A-D2C4-B

##### 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 $1 \le k \le 2n-2$.

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 $1 \le k \le 2n-2$.