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Identifications of Braid Group Representations
by
Thomas Kerler
The Ohio State Univeristy
Coauthors: Craig Jackson (Univeristy of Chicago)

This is a report on results emerging from the masters thesis of Craig Jackson, available at http://www.math.ohio-state.edu/ kerler/papers/craigthesis.ps

In the first part we consider the Krammer-Lawrence representation over Z[q, q^-1, t, t^-1] that has recently been proved faithful by Bigelow. We also consider the N-fold tensor product of the U_xi(sl₂) Verma module with heighets weight lambda (a generic complex number) and consider the R-matrix induced B_N-representation restricted to the subspace of sl₂-weight (N*lambda-4). We show both representations are equivalent if we identify t with xi^lambda and q with xi^-2.

In the second part we look at the stochastic representation of the string link semigroup over Q[t, t^-1] as proposed by Jones and constructed and investigated by Lin et al., which restricts to the unreduced Burau representations on braids. We identify the matrices over Q[t, t^-1] as ratios of two very naturally, skein theoretically defined representations over Z[t, t^-1], thus yielding an efficient way of computation.

Date received: December 12, 2001

Copyright © 2001 by the author(s). The author(s) of this work and the organizers of the conference have granted their consent to include this abstract in Topology Atlas. Document # caip-06.