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 first Lawrence representation is the Burau representation and the second is the 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 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. 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 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