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 to be the mapping class group of a disc with n marked points . The homology group is free abelian of rank n. Moreover, the invariant subspace of (under the action of ) is primitive and infinite cyclic. Let be the projection onto this invariant subspace. Then there is a covering space corresponding to this projection map. Much like in the construction of the Alexander polynomial, consider as a module over the group-ring of covering transformations (a Laurent polynomial ring). As such a -module, is free of rank n − 1. By the basic theory of covering spaces, acts on , and this representation is called the reduced Burau representation.
The unreduced Burau representation has a similar definition, namely one replaces with its (real, oriented) blow-up at the marked points. Then instead of considering one considers the relative homology where is the part of the boundary of corresponding to the blow-up operation together with one point on the disc's boundary. denotes the lift of to . As a -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 and are reduced and unreduced Burau -modules respectively and is the complement to the diagonal subspace (i.e.: , and acts on by the permutation representation.
Relation to the Alexander polynomial
If a knot is the closure of a braid , then the Alexander polynomial is given by where is the reduced Burau representation of the braid .
The first nonfaithful Burau representations are found without the use of computer, using a notion of winding number or contour integration.  A more conceptual understanding  interprets the linking or winding as coming from Poincaré duality in first homology relative to the basepoint of a covering space, and uses the intersection form (traditionally called Squier's Form as Craig Squier was the first to explore its properties). Stephen Bigelow combined computer techniques and the Long-Paton theorem to show that the Burau representation is not faithful for n ≥ 5. 
The Burau representation for n = 2, 3 has been known to be faithful for some time. The faithfulness of the Burau representation when n = 4 is an open problem.
Squier showed that the Burau representation preserves a sesquilinear form. Moreover, when the variable is chosen to be a transcendental unit complex number near it is a positive-definite Hermitian pairing, thus the Burau representation can be thought of as a map into the Unitary group.
- Burau, Werner (1936). "Über Zopfgruppen und gleichsinnig verdrillte Verkettungen". Abh. Math. Sem. Hamburg 11: 179−186.
- J. Moody, The faithfulness question for the Burau representation, Proc. AMS 1993
- D D Long, M Paton, The Burau representation is not faithful for n ≥ 6, Topology 32 (1993)
- Squier, Craig C (1984). "The Burau representation is unitary". Proceedings of the American Mathematical Society 90 (2): 199–202. doi:10.2307/2045338.
- Bigelow, Stephen (1999). "The Burau representation is not faithful for n = 5". Geometry & Topology 3: 397–404. doi:10.2140/gt.1999.3.397.
- S. Bigelow,International Congress of Mathematicians, Beijing, 2002
- V. Turaev, Faithful representations of the braid groups, Bourbaki 1999-2000