A ribbon Hopf algebra approach to the irreducible representations of centralizer algebras: The Brauer, Birman-Wenzl, and type A Iwahori-Hecke algebras

Abstract

We show how the ribbon Hopf algebra structure on the Drinfel'd-Jimbo quantum groups of Types A, B, C, and D can be used to derive formulas giving explicit realizations of the irreducible representations of the Iwahori-Hecke algebras of type A and the Birman-Wenzl algebras. We use this derivation to give explicit realizations of the irreducible representations of the Brauer algebras as well. The derivation is accomplished by way of a combination of techniques from operator algebras, quantum groups, and the theory of 3-manifold invariants. Although our applications are in the cases of the quantum groups of Types A, B, C, and D, most of the aspects of our approach apply in the general setting o..