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Hopf Algebras and Their Actions on Rings
Susan Montgomery, University of Southern California, Los Angeles, CA
A co-publication of the AMS and CBMS.

CBMS Regional Conference Series in Mathematics
1993; 238 pp; softcover
Number: 82
Reprint/Revision History:
reprinted with corrections 1995, third printing 1999
ISBN-10: 0-8218-0738-2
ISBN-13: 978-0-8218-0738-5
List Price: US$34
Member Price: US$27.20
All Individuals: US$27.20
Order Code: CBMS/82
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See also:

Monoidal Functors, Species and Hopf Algebras - Marcelo Aguiar and Swapneel Mahajan

The last ten years have seen a number of significant advances in Hopf algebras. The best known is the introduction of quantum groups, which are Hopf algebras that arose in mathematical physics and now have connections to many areas of mathematics. In addition, several conjectures of Kaplansky have been solved, the most striking of which is a kind of Lagrange's theorem for Hopf algebras. Work on actions of Hopf algebras has unified earlier results on group actions, actions of Lie algebras, and graded algebras. This book brings together many of these recent developments from the viewpoint of the algebraic structure of Hopf algebras and their actions and coactions. Quantum groups are treated as an important example, rather than as an end in themselves. The two introductory chapters review definitions and basic facts; otherwise, most of the material has not previously appeared in book form. Providing an accessible introduction to Hopf algebras, this book would make an excellent graduate textbook for a course in Hopf algebras or an introduction to quantum groups.


Research mathematicians and graduate students.


"I found that Montgomery's book is an excellent outline of all the topics mentioned and can serve as a useful guide in structuring [a course on Hopf algebras and quantum groups] ... there is an excellent bibliography ... the author has performed a highly useful service to the mathematical community."

-- Zentralblatt MATH

"A good guidebook to recent developments in Hopf algebra theory with an emphasis on their actions and coactions on algebras ... Most of the results have not previously appeared in book form ... This book could also be used as a textbook for graduate students ... The author, S. Montgomery, has contributed much to group actions on rings and their generalizations to Hopf algebras (especially finite dimensional)."

-- Bulletin of the AMS

"This book should be very useful as a comprehensive, up to date introduction to the algebraic aspects of Hopf algebras."

-- Monatshefte für Mathematik

Table of Contents

  • Definitions and examples
  • Integrals and semisimplicity
  • Freeness over subalgebras
  • Actions of finite-dimensional Hopf algebras and smash products
  • Coradicals and filtrations
  • Inner actions
  • Crossed products
  • Galois extensions
  • Duality
  • New constructions from quantum groups
  • Some quantum groups
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