Braid groups are linear

By Daan Krammer. 2004

103 citations recorded by the source corpus.

In a previous work [11], the author considered a representation of the braid group p: Bn -- GLm(Z[ql1, t±l]) (m = n(n-1)/2), and proved it to be faithful for n = 4. Bigelow [3] then proved the same representation to be faithful for all n by a beautiful topological argument. The present paper gives a different proof of the faithfulness for all n. We establish a relation between the Charney length in the braid group and exponents of t. A certain Bn-invariant subset of the module is constructed whose properties resemble those of convex cones. We relate line segments in this set with the Thurston normal form of a braid.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/5fcd7283a0027e206cdda9bcee6cb349e94f0bab

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Artin monoids inject in their groups (2001, methodology): “ion 4 that ψis injective. If Γ = A n, then G A n is the braid group on n+ 1 strings, and ψ: G + A n → Gl(V) is equivalent to the representation constructed by Bigelow and Krammer in [Big], [Kra1] and [Kra2]. In this case, V has ﬁnite dimension, and the injectivity of ψimplies the injectivity of the ”
Basic results on braid groups (2010, methodology): “ing open question until it was solved for B4 by Krammer [51], and then for general Bn using topological methods by Bigelow [8]. Shortly after that Krammer generalized his algebraic approach to all Bn [52]. Theorem 5.1. [51, 8, 52] There exists a faithful representation ρ : Bn → GLN(k), where k = Q(p”
Braid groups are linear (2000, methodology): “Recently, Krammer [Kra00] extended his algebraic methods to obtain a new proof of the above theorem as well as some new ways to read information about a braid from its Lawrence-Krammer representation.”