$N$-Dimensional Representations of the Braid Groups $B_{N}$

We study the kernel of the evaluated Burau representation through the braid element $\sigma_i \sigma_{i+1} \sigma_i$. The element is significant as a part of the standard braid relation. We establish the form of this element's image raised to the $n^{th}$ power. Interestingly, the cyclotomic polynomials arise and can be used to define the expression. The main result of this paper is that the Burau representation of the braid group of $n$ strands for $n \geq 3$ is unfaithful a...

In this paper we indicate one method of construction of linear representations of groups and algebras with translation invariant (except, maybe , finite number) defining relationships. As an illustration of this method, we give one approach to the construction of linear representations of braid group and derive some series of such representations. Some invariants of oriented knots and links are constructed. The author is grateful to Yuri Drozd, Sergey Ovsienko and other mem...

These are lecture notes prepared for a minicourse given at the Cimpa Research School "Algebraic and geometric aspects of representation theory", held in Curitiba, Brazil in March 2013. The purpose of the course is to provide an introduction to the study of representations of braid groups. Three general classes of representations of braid groups are considered: homological representations via mapping class groups, monodromy representations via the Knizhnik-Zamolodchikov connec...

In this paper we survey some work on representations of $B_n$ given by the induced action on a homology module of some space. One of these, called the Lawrence-Krammer representation, recently came to prominence when it was shown to be faithful for all $n$. We will outline the methods used, applying them to a closely related representation for which the proof is slightly easier. The main tool is the Blanchfield pairing, a sesquilinear pairing between elements of relative homo...

We characterize unitary representations of braid groups $B_n$ of degree linear in $n$ and finite images of such representations of degree exponential in $n$.

We give a simple topological construction of the Burau representations of the loop braid groups. There are four versions: defined either on the non-extended or extended loop braid groups, and in each case there is an unreduced and a reduced version. Three are not surprising, and one could easily guess the correct matrices to assign to generators. The fourth is more subtle, and does not seem combinatorially obvious, although it is topologically very natural.

The aim of the present note is to enhance groups $G_{n}^{3}$ and to construct new invariants of classical braids. In particular, we construct invariants valued in $G_{N}^{2}$ groups. In groups $G_{n}^{2}$, the identity problem is solved, besides, their structure is much simpler than that of $G_{n}^{3}$. I am grateful to Huyue Yan for pointing out a small mistake in the previous version.

We construct a [(n+1)/2]+1 parameters family of irreducible representations of the Braid group B_3 in arbitrary dimension n\in N, using a q-deformation of the Pascal triangle. This construction extends in particular results by S.P.Humphries [8], who constructed representations of the braid group B_3 in arbitrary dimension using the classical Pascal triangle. E.Ferrand [7] obtained an equivalent representation of B_3 by considering two special operators in the space C^n[X]. Sl...

The Lawrence representation $L_{n,m}$ is a family of homological representation of the braid group $B_n$, which specializes to the reduced Burau and the Lawrence-Krammer representation when $m$ is 1 and 2. In this article we show that the Lawrence representation is faithful for $m \geq 2$.

In his initial paper on braids E.Artin gave a presentation with two generators for an arbitrary braid group. We give analogues of this Artin's presentation for various generalizations of braids.