en>Qetuth: more specific stub type

2012-01-01T10:39:25Z

more specific stub type

New page

In [[mathematics]] the '''Lawrence–Krammer representation''' is a [[group representation|representation]] of the [[braid group]]s. 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.<ref>{{cite arXiv |author=Stephen Bigelow |authorlink= |eprint=math/0204057 |title=The Lawrence–Krammer representation |class= |year=2002 |version=v1 |accessdate=2008-09-08 }}</ref>

== Definition ==

Consider the [[braid group]] <math>B_n</math> to be the [[mapping class group]] of a disc with ''n'' marked points <math>P_n</math>. The Lawrence–Krammer representation is defined as the action of <math>B_n</math> on the homology of a certain [[covering map|covering]] space of the [[configuration space]] <math>C_2 P_n</math>. Specifically, <math>H_1 C_2 P_n \simeq \mathbb Z^{n+1}</math>, and the subspace of <math> H_1 C_2 P_n</math> invariant under the action of <math>B_n</math> is primitive, free and of rank 2. Generators for this invariant subspace are denoted by <math>q, t</math>.

The covering space of <math>C_2 P_n</math> corresponding to the kernel of the projection map

:<math>\pi_1 C_2 P_n \to \mathbb Z^2 \langle q,t \rangle </math>

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

:<math>H_2 \overline{C_2 P_n},</math>

thought of as a

:<math>\mathbb Z\langle t^{\pm},q^{\pm}\rangle</math>-module,

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

== Matrices ==

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

<math>\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.</math>

== Faithfulness ==

Stephen Bigelow and Daan Krammer have independent proofs that the Lawrence–Krammer representation is [[group representation|faithful]].

== Geometry ==

The Lawrence–Krammer representation preserves a non-degenerate [[sesquilinear form]] which is known to be negative-definite Hermitian provided <math>q, t</math> 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 <math>\frac{n(n-1)}{2}</math>-square matrices. Recently it has been shown that the image of the Lawrence–Krammer representation is [[dense set|dense subgroup]] of the [[unitary group]] in this case.

The sesquilinear form has the explicit description:

<math> \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.</math>

== 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.

{{reflist}}

{{DEFAULTSORT:Lawrence-Krammer representation}}
[[Category:Braid groups]]
[[Category:Representation theory]]

Essential subgroup - Revision history

en>Qetuth: more specific stub type