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Some generalizations of quasitriangular Hopf algebras which give rise to invariants of knots and links
by
David E. Radford
University of Illinois at Chicago
Kauffman and the speaker have been interested for a very long time in generalizations of finite-dimensional Hopf algebras which account for invariants of oriented or unoriented 1-1 tangles, knots and links. Kauffman's quantum algebra accounts for the Jones polynomial. We discovered the appropriate analog, called a oriented quantum algebra, for oriented knots and links.
This talk will discuss the theory of oriented quantum algebras and closely related sructures, sketch their theory, and provide examples. There are interesting relationships between quantum algebras and oriented quantum algebras.
The Drinfel'd double of a finite-dimensional Hopf algebra has a natural oriented quantum algebra structure as do finite-dimensional representations of oriented quantum algebras. There is a very interesting oriented quantum algebra structure on the tensor product of an oriented quantum algebra with itself which is motivated by the fact that, as algebras, the Drinfel'd double D(A) of a factorizable Hopf algebra A is the tensor product of A with itself.
The talk will be a survey of joint work with Kauffman, as well as work by the speaker, on oriented quantum algebras.
Date received: February 2, 2005
Copyright © 2005 by the author(s). The author(s) of this work and the organizers of the conference have granted their consent to include this abstract in Topology Atlas. Document # capo-12.