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 to be the mapping class group of a disc with n marked points,. The Lawrence–Krammer representation is defined as the action of on the homology of a certain covering space of the configuration space. Specifically, the first integralhomology group of is isomorphic to, and the subgroup of invariant under the action of is primitive, free abelian, and of rank 2. Generators for this invariant subgroup are denoted by. The coveringspace of corresponding to the kernel of the projection map is called the Lawrence–Krammer cover and is denoted. Diffeomorphisms of act on, thus also on, moreover they lift uniquely to diffeomorphisms of which restrict to the identity on the co-dimension two boundary stratum. The action of on thought of as a is the Lawrence–Krammer representation. The group is known to be a free -module, of rank.
Matrices
Using Bigelow's conventions for the Lawrence–Krammer representation, generators for the group are denoted for. Letting denote the standard Artin generators of the braid group, we obtain the expression:
Faithfulness
and Daan Krammer have given independent proofs that the Lawrence–Krammer representation is faithful.
Geometry
The Lawrence–Krammer representation preserves a non-degeneratesesquilinear form which is known to be negative-definite Hermitian provided are specialized to suitable unit complex numbers. Thus the braid group is a subgroup of the unitary group of square matrices of size. 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:
