Parameters for which the Lawrence–Krammer representation is reducible
We show that the representation, introduced by Lawrence and 
Krammer to show the linearity of the braid group, is generically 
irreducible. However, for some values of its two parameters when 
these are specialized to complex numbers, it becomes reducible. 
We construct a representation of degree n(n−1)/2 of the BMW algebra 
of type A_(n−1). As a representation of the braid group on n strands, 
it is equivalent to the Lawrence–Krammer representation where 
the two parameters of the BMW algebra are related to those 
appearing in the Lawrence–Krammer representation. We give the 
values of the parameters for which the representation is reducible 
and give the proper invariant subspaces in some cases. We use 
this representation to show that for these special values of the 
parameters, the BMW algebra of type A_(n−1) is not semisimple.