The Cyclotomic Birman-Murakami-Wenzl Algebras

TL;DR

This thesis studies the cyclotomic BMW algebras, generalizing BMW algebras with connections to cyclotomic Hecke algebras and knot theory, establishing their algebraic structure and topological realization.

Contribution

It proves the freeness, topological realization, and cellularity of cyclotomic BMW algebras, advancing understanding of their algebraic and topological properties.

Findings

Freeness of rank k^n (2n-1)!!

Topological realization as cylindrical Kauffman Tangle algebra

Proven cellularity of the algebras

Abstract

----- Please see the pdf file for the actual abstract and important remarks, which could not be put here due to the arXiv length restrictions. ----- This thesis presents a study of the cyclotomic BMW (Birman-Murakami-Wenzl) algebras, introduced by Haring-Oldenburg as a generalization of the BMW algebras associated with the cyclotomic Hecke algebras of type G(k,1,n) (also known as Ariki-Koike algebras) and type B knot theory involving affine/cylindrical tangles. They are shown to be free of rank k^n (2n-1)!! and to have a topological realization as a certain cylindrical analogue of the Kauffman Tangle algebra. Furthermore, the cyclotomic BMW algebras are proven to be cellular, in the sense of Graham and Lehrer. This Ph.D. thesis, completed at the University of Sydney, was submitted September 2007 and passed December 2007.