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Zbl 0808.17003
Kassel, Christian
Quantum groups. (English)
[B] Graduate Texts in Mathematics. 155. New York, NY: Springer-Verlag. xii, 531 p. DM 79.00; öS 616.20; sFr. 79.00 (1995). ISBN 0-387-94370-6/hbk

Quantum groups came to the attention of the general mathematical public, perhaps with Drinfeld's paper, at the 1986 International Congress of Mathematicians at Berkeley. They are Hopf algebras which are noncommutative analogues of universal enveloping algebras of Lie algebras or of functions on an algebraic group. They arose in physics through Yang-Baxter equations, but their subsequent development has involved low- dimensional topology, some quite abstract category theory, monodromy of differential equations, and other areas which a priori seem to be unrelated. For some time, there were almost no general books on quantum groups except for a short book by Manin, but in just a short period, many such books have recently appeared. Some of the authors include Chari and Pressley, Fuchs, Lusztig, Montgomery, Shnider and Sternberg, and Turaev. The emphasis varies from physical (as in Fuchs) to the purely algebraic (as in Montgomery).\par The book of Kassel, under review, takes a middle ground. The first two parts stress the algebraic underpinnings, whereas the last two parts treat relations with low-dimensional topology (knots, links, tangles, braids) and with differential equations (those of Knizhnik- Zamolodchikov). The book is very carefully written, and a diligent reader will find it quite readable. The author succeeds admirably in both introducing the reader to the subject as well as indicating some applications in a reasonably substantial treatment. The first two parts would be a good textbook for an introductory graduate course. Montgomery's book could also be used in this way, but it is more of a survey, whereas the book of Kassel is more complete with respect to proofs and details. The reviewer used an earlier French version of the first two parts recently in such a course, and it was very valuable to the students. The book is recommended for mathematicians who want to get an idea of this currently active and popular area as well as for practitioners in the area as a valuable reference book.\par We list the headings of the four parts and the chapters.\par Part One: Quantum SL(2).\par I. Preliminaries, II. Tensor products, III. The language of Hopf algebras, IV. The quantum plane and its symmetries, V. The Lie algebra of SL(2), VI. The quantum enveloping algebra of sl(2), VII. A Hopf algebra structure on $U\sb q( \text{sl} (2))$.\par Part Two: Universal $R$-matrices.\par VIII. The Yang-Baxter equation and (co) braided bialgebras, IX. Drinfeld's quantum double.\par Part Three: Low-dimensional topology and tensor categories.\par X. Knots, links, tangles and braids, XI. Tensor categories, XII. The tangle category, XIII. Braidings, XIV. Duality in tensor categories, XV. Quasi-bialgebras.\par Part Four: Quantum groups and monodromy.\par XVI. Generalities on quantum enveloping algebras, XVII. Drinfeld and Jimbo's quantum enveloping algebras, XVIII. Cohomology and rigidity theorems, XIX. Monodromy of Knizhnik-Zamolodchikov equations, XX. Postlude. A universal knot invariant.
[E.J.Taft (New Brunswick)]

MSC 2000:: *17B37 Quantum groups and related deformations
16W30 Hopf algebras (assoc. rings and algebras)
57M25 Knots and links in the 3-sphere
17-02 Research monographs (nonassoc. rings and algebras)
16-02 Research monographs (assoc. rings and algebras)
57-02 Research monographs (manifolds)
81R50 Quantum groups and related algebraic methods in quantum theory
18D10 Monoidal categories

Keywords: Knizhnik-Zamolodchikov equations; Hopf algebras; quantum groups; low- dimensional topology; quantum enveloping algebra; Hopf algebra; Yang- Baxter equation; braided bialgebras; Drinfeld's quantum double; tensor categories

Cited in: Zbl 1202.58005 Zbl 1208.20041 Zbl 1135.16021 Zbl 1084.17505 Zbl 1084.17504 Zbl 1006.16056 Zbl 0962.16026 Zbl 1034.17004 Zbl 0931.16022 Zbl 0921.16023 Zbl 0928.16028 Zbl 0840.19001 Zbl 0831.18002

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