The Lawrence-Krammer-Bigelow Representations of the Braid Groups via Quantum SL₂

الأصل · EN

We construct representations of the braid groups Bₙ on n strands on free Z[q,q-1,s,s-1]-modules Wₙ,ₗ using generic Verma modules for an integral version of quantum 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).