The Lawrence-Krammer-Bigelow Representations of the Braid Groups via Quantum SL_2
Geometric Topology
2013-06-03 v2 Quantum Algebra
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).
Keywords
Cite
@article{arxiv.0912.2114,
title = {The Lawrence-Krammer-Bigelow Representations of the Braid Groups via Quantum SL_2},
author = {Craig Jackson and Thomas Kerler},
journal= {arXiv preprint arXiv:0912.2114},
year = {2013}
}
Comments
Added a section on the Temperley-Lieb specialization & various minor editorial corrections