On Krammer's Representation of the Braid Group

Given two nonzero complex parameters $l$ and $m$, we construct by the mean of knot theory a matrix representation of size $\chl$ of the BMW algebra of type $A_{n-1}$ with parameters $l$ and $m$ over the field $\Q(l,r)$, where $m=\unsurr-r$. As a representation of the braid group on $n$ strands, it is equivalent to the Lawrence-Krammer representation that was introduced by Lawrence and Krammer to show the linearity of the braid groups. We prove that the Lawrence-Krammer repres...

We determine the Zariski closure of the representations of the braid groups that factorize through the Birman-Wenzl-Murakami algebra, for generic values of the parameters $\alpha,s$. For $\alpha,s$ of modulus 1 and close to 1, we prove that these representations are unitarizable, thus deducing the topological closure of the image when in addition $\alpha,s$ are algebraically independent.

It is known that the Lawrence-Krammer representation of the Artin group of type $A_{n-1}$ based on the two parameters $t$ and $q$ that was used by Krammer and independently by Bigelow to show the linearity of the braid group on $n$ strands is generically irreducible. Here, we recover this result and show further that for some complex specializations of the parameters the representation is reducible. We give all the values of the parameters for which the representation is redu...

In this note, a new class of representations of the braid groups $B_{N}$ is constructed. It is proved that those representations contain three kinds of irreducible representations: the trivial (identity) one, the Burau one, and an $N$-dimensional one. The explicit form of the $N$-dimensional irreducible representation of the braid group $B_{N}$ is given here.

We show that the Lawrence-Krammer representation based on two parameters that was used by Bigelow and independently Krammer to show the linearity of the braid group is generically irreducible, but that when its parameters are specialized to some nonzero complex numbers, the representation is reducible. To do so, we construct a representation of the BMW algebra inside the Lawrence-Krammer space. As a representation of the braid group, this representation is equivalent to the L...

We introduce a reduced form of a Birman-Murakami-Wenzl Algebra associated to the braid group of Coxeter type B and investigate its semisimplicity, Bratteli diagram and Markov trace. Applications in knot theory and physics are outlined.

We give a method to produce representations of the braid group $B_n$ of $n-1$ generators ($n\leq \infty$). Moreover, we give sufficient conditions over a non unitary representation for being of this type. This method produces examples of irreducible representations of finite and infinite dimension.

We show that the Lawrence--Krammer representation is unitary. We explicitly present the non-singular matrix representing the sesquilinear pairing invariant under the action. We show that reversing the orientation of a braid is equivalent to the transposition of its Lawrence--Krammer matrix followed by a certain conjugation. As corollaries it is shown that the characteristic polynomial of the Lawrence--Krammer matrix is invariant under substitution of its variables with their ...

We give an exposition of the work of Bigelow and Krammer who proved that the Artin braid groups are linear.

The cyclotomic Birman-Murakami-Wenzl (or BMW) algebras B_n^k, introduced by R. Haring-Oldenburg, are extensions of the cyclotomic Hecke algebras of Ariki-Koike, in the same way as the BMW algebras are extensions of the Hecke algebras of type A. In this paper we focus on the case n=2, producing a basis of B_2^k and constructing its left regular representation.