Double centralizer properties for the Drinfeld double of the Taft algebras

Abstract: The Drinfeld double of the taft algebra, $D_n$, whose ground field contains $n$-th roots of unity, has a known list of 2-dimensional irreducible modules. For each of such module $V$, we show that there is a well-defined action of the Temperley-Lieb algebra $TL_k$ on the $k$-fold tensor product of $V$, and this action commutes with that of $D_n$. When $V$ is self-dual and when $k \leq 2(n-1)$, we further establish a isomorphism between the centralizer algebra of $D_n$ on $V^{\otimes k}$, and $TL_k$. Our inductive argument uses a rank function on the TL diagrams, which is compatible with the nesting function introduced by Russell-Tymoczko. This is joint work with Georgia Benkart, Rekha Biswal, Ellen Kirkman and Van Nguyen.

combinatoricsquantum algebrarings and algebrasrepresentation theory

Audience: researchers in the topic

OIST representation theory seminar

Series comments: Timings of this seminar may vary from week to week.