Burau representation

In mathematics the Burau representation is a representation of the braid groups, named after and originally studied by the German mathematician Werner Burau during the 1930s. The Burau representation has two common and near-equivalent formulations, the reduced and unreduced Burau representations. In mathematics the Burau representation is a representation of the braid groups, named after and originally studied by the German mathematician Werner Burau during the 1930s. The Burau representation has two common and near-equivalent formulations, the reduced and unreduced Burau representations. Consider the braid group Bn to be the mapping class group of a disc with n marked points Dn. The homology group H1(Dn) is free abelian of rank n. Moreover, the invariant subspace of H1(Dn) (under the action of Bn) is primitive and infinite cyclic. Let π : H1(Dn) → Z be the projection onto this invariant subspace. Then there is a covering space Cn corresponding to this projection map. Much like in the construction of the Alexander polynomial, consider H1(Cn) as a module over the group-ring of covering transformations Z, which is isomorphic to the ring of Laurent polynomials Z. As a Z-module, H1(Cn) is free of rank n − 1. By the basic theory of covering spaces, Bn acts on H1(Cn), and this representation is called the reduced Burau representation. The unreduced Burau representation has a similar definition, namely one replaces Dn with its (real, oriented) blow-up at the marked points. Then instead of considering H1(Cn) one considers the relative homology H1(Cn, Γ) where γ ⊂ Dn is the part of the boundary of Dn corresponding to the blow-up operation together with one point on the disc's boundary. Γ denotes the lift of γ to Cn. As a Z-module this is free of rank n. By the homology long exact sequence of a pair, the Burau representations fit into a short exact sequence where Vr (resp. Vu) is the reduced (resp. unreduced) Burau Bn-module and D ⊂ Zn is the complement to the diagonal subspace, in other words: and Bn acts on Zn by the permutation representation. Let σi denote the standard generators of the braid group Bn. Then the unreduced Burau representation may be given explicitly by mapping for 1 ≤ i ≤ n − 1, where Ik denotes the k × k identity matrix. Likewise, for n ≥ 3 the reduced Burau representation is given by while for n = 2, it maps