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 this talk.
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