The Lawrence-Krammer-Bigelow Representations of the Braid Groups via Quantum SL_2

TL;DR

This paper constructs and analyzes quantum group-based representations of braid groups, demonstrating their faithfulness and irreducibility, and connecting them to known Lawrence-Krammer-Bigelow representations.

Contribution

It introduces new quantum group representations of braid groups and proves their faithfulness and irreducibility, linking them to established representations.

Findings

W_{n,2} are isomorphic to Lawrence-Krammer-Bigelow representations

W_{n,l} are irreducible over Q(q,s)

Constructed explicit bases and isomorphisms

Abstract

We construct representations of the braid groups B_n on n strands on free Z[q,q^-1,s,s^-1]-modules W_{n,l} using generic Verma modules for an integral version of quantum sl_2. We prove that the W_{n,2} are isomorphic to the faithful Lawrence Krammer Bigelow representations of B_n after appropriate identification of parameters of Laurent polynomial rings by constructing explicit integral bases and isomorphism. We also prove that the B_n-representations W_{n,l} are irreducible over the fractional field Q (q,s).