The Cyclotomic Birman-Murakami-Wenzl Algebras and Cylindrical Tangles

Shona Yu

The algebraic definition of the Birman-Murakami-Wenzl (BMW) algebras uses generators and relations originally inspired by the Kauffman link invariant. They are closely connected with the Artin braid group of type A, Iwahori-Hecke algebras of the symmetric group, and may be thought of as a deformation of the Brauer algebras. Geometrically, the BMW algebra is isomorphic to the Kauffman Tangle algebra. These algebras also feature in the theory of quantum groups, statistical mechanics, and even topological quantum field theory.

In view of these relationships between the BMW algebras and several objects of "type A", several authors have since naturally generalized the BMW algebras for other types of Artin groups; for example, Cohen, Gijsbers and Wales introduced and worked extensively on BMW algebras of types D and E.

Motivated by type B knot theory and the cyclotomic Hecke algebras of type G(k,1,n) (aka the Ariki-Koike algebras), Häring-Oldenburg defined the cyclotomic BMW algebras. In this talk, we investigate the structure of these algebras and show they have a diagrammatic interpretation as a certain cylindrical analogue of the Kauffman Tangle algebras. In particular, we provide a basis which may be explicitly described both algebraically and diagrammatically in terms of "cylindrical" tangles. This basis turns out to be cellular, in the sense of Graham and Lehrer.

This talk is a presentation of the results in my Ph.D. thesis, completed recently at the University of Sydney, Australia.

back to EIDMA Seminar Combinatorial Theory announcements