Lawrence–Krammer representation

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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.

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Definition

Consider the braid group B n {\displaystyle B_{n}} to be the mapping class group of a disc with n marked points, P n {\displaystyle P_{n}} . The Lawrence–Krammer representation is defined as the action of B n {\displaystyle B_{n}} on the homology of a certain covering space of the configuration space C 2 P n {\displaystyle C_{2}P_{n}} . Specifically, the first integral homology group of C 2 P n {\displaystyle C_{2}P_{n}} is isomorphic to Z n + 1 {\displaystyle \mathbb {Z} ^{n+1}} , and the subgroup of H 1 ( C 2 P n , Z ) {\displaystyle H_{1}(C_{2}P_{n},\mathbb {Z} )} invariant under the action of B n {\displaystyle B_{n}} is primitive, free abelian, and of rank 2. Generators for this invariant subgroup are denoted by q , t {\displaystyle q,t} .

The covering space of C 2 P n {\displaystyle C_{2}P_{n}} corresponding to the kernel of the projection map

π 1 ( C 2 P n ) → Z 2 ⟨ q , t ⟩ {\displaystyle \pi _{1}(C_{2}P_{n})\to \mathbb {Z} ^{2}\langle q,t\rangle }

is called the Lawrence–Krammer cover and is denoted C 2 P n ¯ {\displaystyle {\overline {C_{2}P_{n}}}} . Diffeomorphisms of P n {\displaystyle P_{n}} act on P n {\displaystyle P_{n}} , thus also on C 2 P n {\displaystyle C_{2}P_{n}} , moreover they lift uniquely to diffeomorphisms of C 2 P n ¯ {\displaystyle {\overline {C_{2}P_{n}}}} which restrict to the identity on the co-dimension two boundary stratum (where both points are on the boundary circle). The action of B n {\displaystyle B_{n}} on

H 2 ( C 2 P n ¯ , Z ) , {\displaystyle H_{2}({\overline {C_{2}P_{n}}},\mathbb {Z} ),}

thought of as a

Z ⟨ t ± , q ± ⟩ {\displaystyle \mathbb {Z} \langle t^{\pm },q^{\pm }\rangle } -module,

is the Lawrence–Krammer representation. The group H 2 ( C 2 P n ¯ , Z ) {\displaystyle H_{2}({\overline {C_{2}P_{n}}},\mathbb {Z} )} is known to be a free Z ⟨ t ± , q ± ⟩ {\displaystyle \mathbb {Z} \langle t^{\pm },q^{\pm }\rangle } -module, of rank n ( n − 1 ) / 2 {\displaystyle n(n-1)/2} .

Matrices

Using Bigelow's conventions for the Lawrence–Krammer representation, generators for the group H 2 ( C 2 P n ¯ , Z ) {\displaystyle H_{2}({\overline {C_{2}P_{n}}},\mathbb {Z} )} are denoted v j , k {\displaystyle v_{j,k}} for 1 ≤ j < k ≤ n {\displaystyle 1\leq j<k\leq n} . Letting σ i {\displaystyle \sigma _{i}} denote the standard Artin generators of the braid group, we obtain the expression:

σ i ⋅ v j , k = { v j , k i ∉ { 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 ≠ k − 1 , q v j , i + ( 1 − q ) v j , k − ( q 2 − q ) t v i , k i = k − 1 ≠ j , v j , k + 1 i = k , − t q 2 v j , k i = j = k − 1. {\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.}

Faithfulness

Stephen Bigelow and Daan Krammer have given 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 {\displaystyle 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 square matrices of size n ( n − 1 ) / 2 {\displaystyle n(n-1)/2} . Recently² 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:

⟨ v i , j , v k , l ⟩ = − ( 1 − t ) ( 1 + q t ) ( q − 1 ) 2 t − 2 q − 3 { − q 2 t 2 ( q − 1 ) i = k < j < l or i < k < j = l − ( q − 1 ) k = i < l < j 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 + q t ) i < k < j < l ( q − 1 ) 2 ( 1 + q t ) k < i < l < j ( 1 − q t ) ( 1 + q 2 t ) k = i , j = l 0 otherwise {\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<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}}\right.}

References

Bigelow, Stephen (2003), "The Lawrence–Krammer representation", Topology and geometry of manifolds, Proc. Sympos. Pure Math., vol. 71, Providence, RI: Amer. Math. Soc., pp. 51–68, MR 2024629 /wiki/Stephen_Bigelow ↩
Budney, Ryan (2005), "On the image of the Lawrence–Krammer representation", Journal of Knot Theory and Its Ramifications, 14 (6): 773–789, arXiv:math/0202246, doi:10.1142/S0218216505004044, MR 2172897, S2CID 14196563 /wiki/Journal_of_Knot_Theory_and_Its_Ramifications ↩

Definition

Matrices

Faithfulness

Geometry

Further reading

References