Essential subgroup

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.

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, $H_1{C}_2{P}_n\simeq {\mathbb{\mathbb{Z}}}^{n+1}$, 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

${\pi }_1{C}_2{P}_n\to {\mathbb{\mathbb{Z}}}^2⟨q,t⟩$

is called the Lawrence–Krammer cover and is denoted $\overline{C_2{P}_n}$. Diffeomorphisms of$P_n$ 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

$\mathbb{\mathbb{Z}}⟨t^{\pm },q^{\pm }⟩$-module,

is the Lawrence–Krammer representation. $H_2\overline{C_2{P}_n}$ is known to be a free $\mathbb{\mathbb{Z}}⟨t^{\pm },q^{\pm }⟩$-module, of rank $(\genfrac{}{}{0ex}{}{n}{2})$.

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 $1\le j<k\le n$. Letting ${\sigma }_i$ denote the standard Artin generators of the braid group, we get the expression:

${\sigma }_i\cdot v_{j,k}=\{\begin{array}{lr}{v}_{j,k}& i\notin \{j-1,j,k-1,k\},\\ q{v}_{i,k}+(q^2-q)v_{i,j}+(1-q)v_{j,k}& i=j-1\\ v_{j+1,k}& i=j\ne k-1,\\ q{v}_{j,i}+(1-q)v_{j,k}-(q^2-q)t{v}_{i,k}& i=k-1\ne j,\\ v_{j,k+1}& i=k,\\ -t{q}^2{v}_{j,k}& i=j=k-1.\end{array}$

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:

$⟨v_{i,j},v_{k,l}⟩=-(1-t)(1+qt)(q-1{)}^2{t}^{-2}{q}^{-3}\{\begin{array}{lr}-q^2{t}^2(q-1)& i=k<j<l\text{ or }i<k<j=l\\ -(q-1)& k=i<l<j\text{ or }k<i<j=l\\ t(q-1)& i<j=k<l\\ q^{2}t(q-1)& k<l=i<j\\ -t(q-1{)}^2(1+qt)& i<k<j<l\\ (q-1{)}^2(1+qt)& k<i<l<j\\ (1-qt)(1+q^{2}t)& k=i,j=l\\ 0& \text{otherwise}\\ \end{array}$

References

• S. Bigelow, Braid groups are linear, J. Amer. Math. Soc. 14 (2001), 471-486.
• S. Bigelow, The Lawrence–Krammer representation, Topology and geometry of manifolds, Proc. Sympos. Pure Math., 71 (2003)
• R. Budney, On the image of the Lawrence–Krammer representation, J Knot. Th. Ram. (2005)
• D. Krammer, Braid groups are linear, Ann. Math. 155 (2002), 131-156.
• L. Paoluzzi and L. Paris, A note on the Lawrence-Krammer-Bigelow representation, Alg. Geom. Topology 2 (2002), 499-518.

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