#43: New directions in Hopf algebras
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# MSRI Publications -- Volume 43

## New directions in Hopf algebras

### Edited by
Susan Montgomery and Hans-Jürgen Schneider

Hopf algebras have important connections to quantum theory, Lie
algebras, knot and braid theory, operator algebras, and other
areas of physics and mathematics. They have been intensely studied in
the last decade; in particular, the solution of a number of
conjectures of Kaplansky from the 1970s has led to progress on the
classification of semisimple Hopf algebras and on the structure of
pointed Hopf algebras. There has been much progress also on actions
and coactions of Hopf algebras and on Hopf Galois extensions. Many
new methods have been used for these results: modular and
braided categories, representation theory, algebraic geometry, and Lie
methods such as Cartan matrices.
The contributors to this volume of expository papers were participants
in the Hopf Algebras Workshop held at MSRI as part of the 1999--2000
Year on Noncommutative Algebra. Together the papers give a clear
picture of the current trends in this active field, with a focus on
what is likely to be important in future research.

Among the topics covered are results toward the classification of
finite-dimensional Hopf algebras (semisimple and non-semisimple),
as well as what is known about the
extension theory of Hopf algebras. Some papers consider Hopf versions
of classical topics, such as the Brauer group, while others are closer
to recent work in quantum groups. The book also explores the
connections and applications of Hopf algebras to other fields.