论文标题
TAFT代数的Drinfeld双重的张量表示
Tensor Representations for the Drinfeld Double of the Taft Algebra
论文作者
论文摘要
在代数封闭的字段$ \ mathbb k $的特征零，drinfeld double $ d_n $是使用unity $ q \ in \ mathbb k $ in \ n \ geq 2 $ for $ n \ geq 2 $的原始$ n $ th root定义的taft代数的d_n $。 Kauffman和Radford表明$ d_n $在且仅当$ n $奇怪并且功能区元素是唯一的时具有色带元素；但是，没有对此元素的明确描述。在这项工作中，我们明确确定$ d_n $的功能区元素。对于任何$ n \ geq 2 $，我们使用$ d_n $的r-matrix构建temperley-lieb algebra $ \ mathsf {tl} _k（ξ）$，用$ξ= - （q^{\ frac {\ frac {1}}}}}+q^frac} $ k $ -fold Tensor Power $ v^{\ otimes k} $的任何二维简单$ d_n $ -module $ v $。众所周知，此操作是任意$ k \ geq 1 $的忠诚。我们表明，$ \ Mathsf {tl} _k（ξ）$是centralizer algebra $ \ text {end} _ {d_n}（v^{\ otimes k}）$ for $ 1 \ le K \ le 2n-2 $。
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 $\mathsf{TL}_k(ξ)$ with $ξ= -(q^{\frac{1}{2}}+q^{-\frac{1}{2}})$ 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 $\mathsf{TL}_k(ξ)$ is isomorphic to the centralizer algebra $\text{End}_{D_n}(V^{\otimes k})$ for $1 \le k \le 2n-2$.
