In mathematics the Lawrence–Krammer representation is a representation of the braid groups. It fits into a family of representations called the Lawrence representations. The first Lawrence representation is the Burau representation and the second is the Lawrence–Krammer representation. The Lawrence–Krammer representation is named after Ruth Lawrence and Daan Krammer.

Definition

Consider the braid group B_n to be the mapping class group of a disc with n marked points, P_n. The Lawrence–Krammer representation is defined as the action of B_n on the homology of a certain covering space of the configuration space C_2 P_n. Specifically, the first integral homology group of C_2 P_n is isomorphic to, and the subgroup of invariant under the action of B_n is primitive, free abelian, and of rank 2. Generators for this invariant subgroup are denoted by q, t. The covering space of C_2 P_n corresponding to the kernel of the projection map is called the Lawrence–Krammer cover and is denoted. Diffeomorphisms of P_n act on P_n, thus also on C_2 P_n, moreover they lift uniquely to diffeomorphisms of which restrict to the identity on the co-dimension two boundary stratum (where both points are on the boundary circle). The action of B_n on thought of as a is the Lawrence–Krammer representation. The group is known to be a free -module, of rank n (n-1)/ 2.

Matrices

Using Bigelow's conventions for the Lawrence–Krammer representation, generators for the group are denoted v_{j,k} for. Letting \sigma_i denote the standard Artin generators of the braid group, we obtain the expression:

Faithfulness

Stephen Bigelow and Daan Krammer have given independent proofs that the Lawrence–Krammer representation is faithful.

Geometry

The Lawrence–Krammer representation preserves a non-degenerate sesquilinear form which is known to be negative-definite Hermitian provided q, t are specialized to suitable unit complex numbers (q near 1 and t near i). Thus the braid group is a subgroup of the unitary group of square matrices of size n(n-1)/2. Recently it has been shown that the image of the Lawrence–Krammer representation is a dense subgroup of the unitary group in this case. The sesquilinear form has the explicit description: