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Hopf algebras and their actions on rings. Expanded version of ten lectures given at the CBMS Conference on Hopf algebras and their actions on rings, which took place at DePaul University in Chicago, USA, August 10-14, 1992. (English) Zbl 0793.16029
Regional Conference Series in Mathematics. 82. Providence, RI: American Mathematical Society (AMS). ix, 238 p. (1993).
These lecture notes are an expanded version of ten lectures given by the author at a conference in Chicago in 1992. The point of view is the algebraic structure of Hopf algebras and their actions and coactions. There has been an increased interest in Hopf algebras in recent years because of their appearance in statistical mechanics as quantum groups. In these notes, quantum groups are treated as important examples, but this still serves as a useful introduction to quantum groups. Hopf algebra ideas which arose through quantum groups are discussed in the last chapter (e.g. quasitriangular Hopf algebras, Drinfeld doubles).
The appearance of these notes, together with the recent appearance of the notes of Ch. Kassel, based on courses he taught at the University of Strasbourg, are quite timely. Montgomery’s notes also grew partly out of courses she taught at the University of Southern California. The last books on Hopf algebras were those by Sweedler in 1969 and Abe in 1980 (English version). Of the ten chapters in the notes under review, Chapters 1 and 2, and parts of Chapters 5 and 9 appear in those earlier books. The remaining chapters cover the Nichols-Zoeller freeness theorem, recent results on smash products and invariants, inner actions and Skolem-Noether type theorems, cleft extensions and crossed products, Galois extensions, and material (some of which is mentioned above) related to quantum groups.
The reviewer used these notes as an aid in teaching a course on Hopf algebras and quantum groups in the fall semester of 1993. They were used basically as a reference, along with the notes of Kassel, the books of Sweedler and Abe, and the reviewer’s personal notes on quantum groups. Kassel’s notes are somewhat more oriented towards quantum groups. I found that Montgomery’s book is an excellent outline of all the topics mentioned, and can serve as a useful guide in structuring such a course. Many results are quoted without proof, but there is an excellent bibliography, and references are given for all unproved results. Thus one can use the other notes and books, together with some original articles, to put everything together. There have been some recent books on quantum groups, but the field is evolving rapidly, so that if one wants to teach a course on Hopf algebras, including examples of quantum groups and a few basic ideas arising from them, these lectures, supplemented as described above, represent a good way to organize such a course. After the impact of quantum groups is finally understood, it would then perhaps be appropriate for someone to write a definitive treatise on the algebraic theory of Hopf algebras. In the meantime, the author has performed a highly useful service to the mathematical community.

16W30 Hopf algebras (associative rings and algebras) (MSC2000)
16S40 Smash products of general Hopf actions
16-02 Research exposition (monographs, survey articles) pertaining to associative rings and algebras