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

We construct representations of the braid groups Bn on n strands on free Z[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(q,s).