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The motivation behind the definition of the Birman-Murakami-Wenzl (BMW)
algebras may be traced back to an important problem in knot theory:
namely, that of classifying knots (and links) up to isotopy, which leads
to the study of link invariants.
The algebraic definition of the BMW algebras uses generators and relations
originally inspired by the Kauffman link invariant. They are intimately
connected with the Artin braid group of type A, Iwahori-Hecke algebras of
type A (the symmetric group), and with many diagram algebras (algebras with
basis a given set of diagrams where multiplication is described by a
simple diagram calculus).
Geometrically, the BMW algebra is isomorphic to the Kauffman tangle
algebra. The representations and the cellularity of the BMW algebra have
now been extensively studied in the literature. These algebras also
feature in the theory of quantum groups, statistical mechanics, and
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 algberas for other types of Artin groups. Motivated by knot theory associated with the Artin braid group of type B, Häring-Oldenburg introduced the cyclotomic BMW algebras as a generalization of the BMW algebras associated with the Ariki-Koike algebras, aka the cyclotomic Hecke algebras of type G(k,1,n). 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. |
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