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

Thomas Kerler (Ohio State University)

Abstract: Around 2001 Krammer and Bigelow proved the long standing conjecture that the braid groups are linear.

The representation of B_n they prove to be faithful acts on the second homology group of a certain covering space of the 2-point configuration space of an n-punctured disc. The covering group is free of rank 2 so that the module is naturally over the ring of Laurent polynomials in two variables.

Another family of representations of B_n is constructed using quantum-sl_2, which is a one-parameter deformation of the universal enveloping algebra of sl_2. The algebra is admits a quasi-triangular R-matrix which can be used to represent B_n on the n-fold tensor product of a Verma module with generic highest weight.

We prove that a subspace of the latter representation is isomorphic to the LKB representation where the two parameters corresponding to covering transformations are identified with the deformation parameter of quantum-sl_2 and the generic highest weight of the Verma module respectively. We also show irreducibility of this representation over the fraction field of the ring of Laurent polynomials.

Joint work with Craig Jackson.