  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 Lawrence-Krammer representation, where the
two parameters of the algebra are related to the parameters of the
Lawrence-Krammer representation. We give all the complex values of the
parameters for which the representation is reducible and describe the invariant
subspaces in some cases. We show that for these values of the parameters and
other values, the BMW algebra is not semisimple.
