Quantum Groups-The Functorial Side
by
Bodo Pareigis
Mathematisches Institut der Universität München

Through their multiplication and comultiplication Quantum groups or Hopf algebras carry structural elements that correspond to the structural elements of monoidal categories (tensor categories) and monoidal functors. We will pursue these connexions.

We start with the reconstruction of an algebra A from its module category A-Mod and the underlying functor U:A-Mod --> \Vek to vector spaces. On this we build a dictionary.

A bialgebra structure on A corresponds to a monoidal structure on A-Mod and U. A Hopf algebra structure on A corresponds to duals of (finite-dimensional) A-modules. The most interesting developments are found in the correspondence between (quasi-)triangular structures on A and symmetries (braidings) on a monoidal category A-Mod. Special cases are the categories of super vector spaces, color vector spaces and much more general constructions.

The center construction on a category leads to the category of Yetter-Drinfeld modules and Drinfeld's famous ''Drinfeld Double'', a quasitriangular Hopf algebra whose modules form a braided monoidal category in an almost universal way.

A generalization of the definition of Yetter Drinfeld modules leads to a categorical riddle: an example of a universal-couniversal problem, that is defined by a simultaneous unit and counit.

Date received: June 27, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caeq-54.