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Mathematics, Rings and Algebras, Representation Theory
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 $1 \le k \le 2n-2$.
