  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 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.
