Irreducibility of the Lawrence-Krammer representation of the BMW algebra of type $A_{n-1}$, PhD thesis California Institute of Technology 2008

Given two nonzero complex parameters $l$ and $m$, we construct by the mean of knot theory a matrix representation of size $\text{chl}$ of the BMW algebra of type $A_{n-1}$ with parameters $l$ and $m$ over the field $\text{Q}(l,r)$, where $m=\text{unsurr}-r$. 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 representation is generically irreducible, but that for some values of the parameters $l$ and $r$, it becomes reducible. In particular, we show that for these values of the parameters $l$ and $r$, the BMW algebra is not semisimple. When the representation is reducible, the action on a proper invariant subspace of the Lawrence-Krammer space must be a Hecke algebra action. It allows us to describe the invariant subspaces when the representation is reducible.