Drinfeld doubles of the $n$-rank Taft algebras and a generalization of the Jones polynomial

Ge Feng, Naihong Hu and Yunnan Li

Vol. 312 (2021), No. 2, 421–456

DOI: 10.2140/pjm.2021.312.421

Abstract

We describe the Drinfeld double structure of the $n$-rank Taft algebra and all of its simple modules, and then endow its $R$-matrices with an application to knot invariants. The knot invariant we get is a generalization of the Jones polynomial, in particular, it coincides with the Jones polynomial in the rank $1$ case, while in the rank $2$ case, it is the one-parameter specialization of the two-parameter unframed Dubrovnik polynomial, and in higher rank case it is the composite ($n$-power) of the Jones polynomial.

Keywords

$n$-rank Taft algebra, Drinfeld double, knot invariants, a generalization of the Jones polynomial

Mathematical Subject Classification

Primary: 16T05, 16T25, 17B37, 81R50

Secondary: 57K14, 57K16

Milestones

Received: 23 November 2020

Revised: 8 March 2021

Accepted: 23 March 2021

Published: 31 August 2021