Lawrence–Krammer representation

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 1st Lawrence representation is the Burau representation and the 2nd is the Lawrence–Krammer representation.

The Lawrence–Krammer representation is named after Ruth Lawrence and Daan Krammer.[1]

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, H_1 C_2 P_n \simeq \mathbb Z^{n+1}, and the subspace of  H_1 C_2 P_n invariant under the action of B_n is primitive, free and of rank 2. Generators for this invariant subspace are denoted by q, t.

The covering space of C_2 P_n corresponding to the kernel of the projection map

\pi_1 C_2 P_n \to \mathbb Z^2 \langle q,t \rangle

is called the Lawrence–Krammer cover and is denoted \overline{C_2 P_n}. Diffeomorphisms ofP_n act on P_n, thus also on C_2 P_n, moreover they lift uniquely to diffeomorphisms of \overline{C_2 P_n} which restrict to identity on the co-dimension two boundary stratum (where both points are on the boundary circle). The action of B_n on

H_2 \overline{C_2 P_n},

thought of as a

\mathbb Z\langle t^{\pm},q^{\pm}\rangle-module,

is the Lawrence–Krammer representation. H_2 \overline{C_2 P_n} is known to be a free \mathbb Z\langle t^{\pm},q^{\pm}\rangle-module, of rank n \choose 2.

Matrices

Using Bigelow's conventions for the Lawrence–Krammer representation, generators for H_2 \overline{C_2 P_n} are denoted v_{j,k} for 1 \leq j < k \leq n. Letting \sigma_i denote the standard Artin generators of the braid group, we get the expression:

\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^2v_{j,k} & i=j=k-1.
\end{array}
\right.

Faithfulness

Stephen Bigelow and Daan Krammer have 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 \frac{n(n-1)}{2}-square matrices. Recently it has been shown that the image of the Lawrence–Krammer representation is dense subgroup of the unitary group in this case.

The sesquilinear form has the explicit description:

 \langle v_{i,j}, v_{k,l}\rangle = -(1-t)(1+qt)(q-1)^2t^{-2}q^{-3}
\left\{
\begin{array}{lr}
-q^2t^2(q-1) & i=k<j<l \text{ or } i<k<j=l \\
-(q-1) & k=i<l<j \text{ or } k<i<j=l \\
t(q-1) & i<j=k<l \\
q^2t(q-1) & k<l=i<j \\
-t(q-1)^2(1+qt) & i<k<j<l \\
(q-1)^2(1+qt) & k<i<l<j \\
(1-qt)(1+q^2t) & k=i, j=l \\
0 & \text{otherwise} \\
\end{array}
\right.

References

  1. Stephen Bigelow (2002). "The Lawrence–Krammer–Bigelow representation". arXiv:math/0204057v1.

Further reading

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