Parameters for which the Lawrence-Krammer representation is reducible

In this paper, we classify the finite dimensional irreducible modules for affine BMW algebra over an algebraically closed field with arbitrary characteristic.

We introduce a Brauer type algebra $B_G (\Upsilon) $ associated with every pseudo reflection group and every Coxeter group $G$. When $G$ is a Coxeter group of simply-laced type we show $B_G (\Upsilon)$ is isomorphic to the generalized Brauer algebra of simply-laced type introduced by Cohen, Gijsbers and Wales ({\it J. Algebra}, {\bf 280} (2005), 107-153). We also prove $B_G (\Upsilon)$ has a cellular structure and be semisimple for generic parameters when $G$ is a dihedral gr...

We characterize all simple unitarizable representations of the braid group $B_3$ on complex vector spaces of dimension $d \leq 5$. In particular, we prove that if $\sigma_1$ and $\sigma_2$ denote the two generating twists of $B_3$, then a simple representation $\rho:B_3 \to \gl(V)$ (for $\dim V \leq 5$) is unitarizable if and only if the eigenvalues $\lambda_1, \lambda_2, ..., \lambda_d$ of $\rho(\sigma_1)$ are distinct, satisfy $|\lambda_i|=1$ and $\mu^{(d)}_{1i} > 0$ for $2...

Braid groups are an important and flexible tool used in several areas of science, such as Knot Theory (Alexander's theorem), Mathematical Physics (Yang-Baxter's equation) and Algebraic Geometry (monodromy invariants). In this note we will focus on their algebraic-geometric aspects, explaining how the representation theory of higher genus braid groups can be used to produce interesting examples of projective surfaces defined over the field of complex numbers.

We classify the finite dimensional irreducible representations of affine Hecke algebras of type B_2 with unequal parameters.

We construct a 3-variable enrichment of the Lawrence-Krammer-Bigelow (LKB) representation of the braid groups, which is the limit of a pro-nilpotent tower of representations having the original LKB representation as its bottom layer. We also construct analogous pro-nilpotent towers of representations of surface braid groups and loop braid groups.

We consider Albeverio- Rabanovich linear representation $\pi$ of the braid group $B_3$. After specializing the indeterminates used in defining the representation to non-zero complex numbers, we prove that the restriction of $\pi$ to the pure braid group $P_3$ of dimension three is irreducible.

Using knot theory, we construct a linear representation of the CGW algebra of type $D_n$. This representation has degree $n^2-n$, the number of positive roots of a root system of type $D_n$. We show that the representation is generically irreducible, but that when the parameters of the algebra are related in a certain way, it becomes reducible. As a representation of the Artin group of type $D_n$, this representation is equivalent to the faithful linear representation of Cohe...

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

In this paper we determine the complex generic representation theory of the Juyumaya algebra. We show that a certain specialization of this algebra is isomorphic to the small ramified partition algebra, introduced by P. Martin.