The Lawrence-Krammer-Bigelow representations of the braid groups via Uq(sl2)

Abstract

We construct representations of the braid groups Bn on n strands on free Z{double-struck}[q±1,s±1]-modules Wn,l using generic Verma modules for an integral version of Uq(sl2). We prove that the Wn,2 are isomorphic to the faithful Lawrence-Krammer-Bigelow representations of Bn after appropriate identification of parameters of Laurent polynomial rings by constructing explicit integral bases and isomorphism. We also prove that the Bn-representations Wn,l are irreducible over the fractional field Q{double-struck}(q,s). © 2011 Elsevier Inc.