In this talk we discuss a method to obtain finite dimensional
representations of the braid group of type B via centralizer
algebras of quantum groups. Because of the relationship of quantum
groups to Lie algebras, we are able to use combinatorics of the
theory of weights and their correspondence to Young diagrams to
label the irreducible representations.
Through this method, for example, we obtain specializations of the
Hecke algebra of type B, Ariki-Koike algebras (also known as
cyclotomic Hecke algebras) and the q-Brauer algebra of type B (also
known as the B-BMW algebra). These algebras will be defined in
One application of using the method, described in this talk, is
that it allows us to define a trace on these algebras, called the
Markov trace. This trace is important in determining when the
algebras in the previous paragraph are semisimple.
part of this talk is joint work with H. Wenzl