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.
[1] 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 integral homology group of
invariant under the action of
is primitive, free abelian, and of rank 2.
Generators for this invariant subgroup are denoted by
The covering space of
corresponding to the kernel of the projection map is called the Lawrence–Krammer cover and is denoted
Diffeomorphisms of
, 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).
on thought of as a is the Lawrence–Krammer representation.
is known to be a free
-module, of rank
Using Bigelow's conventions for the Lawrence–Krammer representation, generators for the group
denote the standard Artin generators of the braid group, we obtain the expression:
{\displaystyle \sigma _{i}\cdot v_{j,k}=\left\{{\begin{array}{lr}v_{j,k}&i\notin \{j-1,j,k-1,k\},\\qv_{i,k}+(q^{2}-q)v_{i,j}+(1-q)v_{j,k}&i=j-1\\v_{j+1,k}&i=j\neq k-1,\\qv_{j,i}+(1-q)v_{j,k}-(q^{2}-q)tv_{i,k}&i=k-1\neq j,\\v_{j,k+1}&i=k,\\-tq^{2}v_{j,k}&i=j=k-1.\end{array}}\right.}
Stephen Bigelow and Daan Krammer have given independent proofs that the Lawrence–Krammer representation is faithful.
The Lawrence–Krammer representation preserves a non-degenerate sesquilinear form which is known to be negative-definite Hermitian provided
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
Recently[2] 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:
{\displaystyle \langle v_{i,j},v_{k,l}\rangle =-(1-t)(1+qt)(q-1)^{2}t^{-2}q^{-3}\left\{{\begin{array}{lr}-q^{2}t^{2}(q-1)&i=k