**Rosa Orellana**

(Dartmouth U)

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