zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Hopf algebras. An introduction. (English) Zbl 0962.16026
Pure and Applied Mathematics, Marcel Dekker. 235. New York, NY: Marcel Dekker. ix, 401 p. (2001).

Hopf algebras arose in algebraic topology in the work of Heinz Hopf. The paper of J. W. Milnor and J. C. Moore [Ann. Math., II. Ser. 81, 211-264 (1965; Zbl 0163.28202)] on graded Hopf algebras perhaps was the first to attract the attention of algebraists. The first book on the purely algebraic aspects (for the most part ungraded) was that of M. E. Sweedler [Hopf Algebras, Benjamin, New York (1969; Zbl 0194.32901)]. Subsequent books of this nature were those of E. Abe [Hopf Algebras, Cambridge Univ. Press (1980; Zbl 0476.16008)] and S. Montgomery [Hopf Algebras and their Actions on Rings, Reg. Conf. Ser. Math. 82, Am. Math. Soc., Providence RI (1993; Zbl 0793.16029)]. The subject received a huge impetus with the discovery by physicists and mathematicians starting about the mid 1980’s of quantum groups, which are certain noncommutative and noncommutative Hopf algebras. Sweedler’s book is somewhat out of date and Abe’s book is written in a style which is difficult to access, especially for students. The book of Montgomery is an excellent outline, but does not have complete proofs and contains very little directly on quantum groups. Thus there is a need for an introductory book which could be used in a graduate course on Hopf algebras, and the book under review is hence a timely addition to the literature. It can serve as a textbook on the algebraic theory of Hopf algebras. It does not deal with quantum groups, e.g., the quantum Yang-Baxter equation, so that if the instructor wishes to include some quantum groups, he or she would have to add supplementary material. A book which could serve as a textbook for a course on quantum groups is that of C. Kassel [Quantum Groups, Graduate Texts Math. 155, Springer-Verlag (1995; Zbl 0808.17003)].

The titles of the seven chapters are: 1. Algebras and coalgebras. 2. Comodules. 3. Special classes of coalgebras. 4. Bialgebras and Hopf algebras. 5. Integrals. 6. Actions and coaction of Hopf algebras. 7. Finite-dimensional Hopf algebras. There are two appendices on category theory language and on C-groups and C-cogroups.

The presentation is partly categorical and partly concrete algebra, including, for example some Hopf Galois theory and some classification results for finite-dimensional Hopf algebras. Various classes of coalgebras are studied, and investigated for Hopf algebras, with applications given to integrals, Hopf actions and Galois extensions. The idea of duality is extensively used. Some fundamental theorems for finite-dimensional Hopf algebras which are given complete discussions include the Nichols-Zoeller theorem on Hopf subalgebras, the Taft-Wilson theorem on pointed Hopf algebras and the Kac-Zhu theorem on Hopf algebras of prime dimension.

This book is ideal as an up-to-date introduction to the algebraic theory of Hopf algebras. It can be used as a textbook for a one or two semester course. It has exercises of various levels of difficulty scattered in the text, with solutions at the end of each chapter. There are bibliographical notes at the end of each chapter, as well as a bibliography at the end of the book. If such a course wanted to give roughly equal weight to Hopf algebras and to quantum groups, the book under review could be used together with the one of Kassel mentioned above.


MSC:
16W30Hopf algebras (assoc. rings and algebras) (MSC2000)
16-02Research monographs (associative rings and algebras)