## A Ribbon Hopf Algebra Approach to the Irreducible Representations of Centralizer Algebras: The Brauer, Birman-Wenzl, and Type A Iwahori-Hecke Algebras

Arun Ram

Department of Mathematics and Statistics

University of Melbourne

Parkville, VIC 3010 Australia

aram@unimelb.edu.au

Last update: 7 March 2014

## 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 of ribbon Hopf algebras.

## Notes and References

This is a typed version of the paper *A Ribbon Hopf Algebra Approach to the Irreducible Representations of Centralizer Algebras: The Brauer, Birman-Wenzl, and Type A Iwahori-Hecke Algebras* by Robert ${\text{Leduc}}^{*}$ and Arun ${\text{Ram}}^{\u2020}\text{.}$

The paper was received June 24, 1994; accepted September 12, 1994.

${}^{*}$Supported in part by National Science Foundation Grant DMS-9300523 to the University of Wisconsin.

${}^{\u2020}$Supported in part by National Science Foundation Postdoctoral Fellowship DMS-9107863.

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