Braid group action on projective quantum $\mathfrak{sl}(2)$ modules

TL;DR

This paper constructs a new family of braid group representations using quantum algebra techniques, extending known representations at roots of unity and proving faithfulness of the action.

Contribution

It introduces a novel family of braid group representations derived from the quantum $ ext{sl}(2)$ algebra acting on projective modules, extending Lawrence representations.

Findings

The new representations extend Lawrence representations at roots of unity.

The braid group action is faithful despite finite order center.

The construction uses the R-matrix of the quasitriangular quantum algebra.

Abstract

We define a family of the braid group representations via the action of the $R$-matrix (of the quasitriangular extension) of the restricted quantum $\mathfrak{sl}(2)$ on a tensor power of a simple projective module. This family is an extension of the Lawrence representation specialized at roots of unity. Although the center of the braid group has finite order on the specialized Laurence representations, this action is faithful for our extension.