Tannaka Duality and Quantum Groups

Abstract:  This was the first of three
seminars on my recently completed Ph.D.
thesis: Categories of Representations of
Balanced Coalgebroids.  It provided
motivational material from  the theory of
Tannaka duality and quantum groups [JS91].

First, Pontrjagin duality, and classical
Tannaka Duality were outlined.  Then the
Tannaka duality of [JS91] was described.
Three theorems from [JS91] were highlighted:
First that a coalgebra can be reconstructed
from its category of representations;
secondly, categories equipped with a
functor into the category of finite dimensional
vector spaces that are  equivalent to the
category of representations of some coalgebra
can be characterized in elementary terms;
and thirdly, that extra structure
on a coalgebra can be reconstructed from the
coresponding structure on its category
of representations.  The defintions
of tortile Hopf algebra and tortile
monoidal category were given, and it was observed
that the classical quantum groups are
examples of the former.

Bibliograpy.

[JS91]  A. Joyal and R. Street, An introduction
to Tannaka dualty and quantum groups. Springer
Lecture Notes in Mathematics, Vol 1488, 1991,
pages 411-492.