A basis for the Birman-Wenzl algebra

Abstract

An explicit isomorphism is constructed between the Birman-Wenzl algebra, defined algebraically by J. Birman and H. Wenzl using generators and relations, and the Kauffman algebra, constructed geometrically by H. R. Morton and P. Traczyk in terms of tangles. The isomorphism is obtained by constructing an explicit basis in the Birman-Wenzl algebra, analogous to a basis previously constructed for the Kauffman algebra using 'Brauer connectors'. The geometric isotopy arguments for the Kauffman algebra are systematically replaced by algebraic versions using the Birman-Wenzl relations.

This is a lightly edited version of an article written in 1989 but never fully completed. It was originally intended as a joint paper with A. J.Wassermann. 33 pages, 20 figures

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• Irreducibility of the Lawrence-Krammer representation of the BMW algebra of type $A_{n-1}$, PhD thesis California Institute of Technology 2008
• The Burman-Murakami-Wenzl algebras of type En
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• Tangles of type $E_n$ and a reducibility criterion for the Cohen-Wales representation of the Artin group of type $E_6$