Vol. 312, No. 2, 2021

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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
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
Authors
Ge Feng
School of Mathematical Sciences and Shanghai Key Laboratory of PMMP
East China Normal University
Shanghai 200241
China
Shanghai Center for Mathematical Sciences
Fudan University
Shanghai 200433
China
Naihong Hu
School of Mathematical Sciences and Shanghai Key Laboratory of PMMP
East China Normal University
Shanghai 200241
China
Yunnan Li
School of Mathematics and Information Science
Guangzhou University
Guangzhou 510006
China