On Krammer’s representation of the braid group. (English) Zbl 1042.20023

Zinno, Matthew G.

Math. Ann. 321, No. 1, 197-211 (2001).

Summary: A connection is made between the Krammer representation and the Birman-Murakami-Wenzl algebra. Inspired by a dimension argument, a basis is found for a certain irrep of the algebra, and relations which generate the matrices are found. Following a rescaling and change of parameters, the matrices are found to be identical to those of the Krammer representation. The two representations are thus the same, proving the irreducibility of one and the faithfulness of the other.

MSC:

20F36	Braid groups; Artin groups
57M25	Knots and links in the \(3\)-sphere (MSC2010)

Keywords:

Krammer representation; Birman-Murakami-Wenzl algebras; irreducible representations; faithful representations