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**Quantum groups.**
*(English)*
Zbl 0808.17003

Graduate Texts in Mathematics. 155. New York, NY: Springer-Verlag. xii, 531 p. (1995).

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).

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.

We list the headings of the four parts and the chapters.

Part One: Quantum SL(2).

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 \(U_ q( \text{sl} (2))\).

Part Two: Universal \(R\)-matrices.

VIII. The Yang-Baxter equation and (co) braided bialgebras, IX. Drinfeld’s quantum double.

Part Three: Low-dimensional topology and tensor categories.

X. Knots, links, tangles and braids, XI. Tensor categories, XII. The tangle category, XIII. Braidings, XIV. Duality in tensor categories, XV. Quasi-bialgebras.

Part Four: Quantum groups and monodromy.

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.

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.

We list the headings of the four parts and the chapters.

Part One: Quantum SL(2).

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 \(U_ q( \text{sl} (2))\).

Part Two: Universal \(R\)-matrices.

VIII. The Yang-Baxter equation and (co) braided bialgebras, IX. Drinfeld’s quantum double.

Part Three: Low-dimensional topology and tensor categories.

X. Knots, links, tangles and braids, XI. Tensor categories, XII. The tangle category, XIII. Braidings, XIV. Duality in tensor categories, XV. Quasi-bialgebras.

Part Four: Quantum groups and monodromy.

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.

Reviewer: E.J.Taft (New Brunswick)

### MSC:

17B37 | Quantum groups (quantized enveloping algebras) and related deformations |

16W30 | Hopf algebras (associative rings and algebras) (MSC2000) |

57M25 | Knots and links in the \(3\)-sphere (MSC2010) |

17-02 | Research exposition (monographs, survey articles) pertaining to nonassociative rings and algebras |

16-02 | Research exposition (monographs, survey articles) pertaining to associative rings and algebras |

57-02 | Research exposition (monographs, survey articles) pertaining to manifolds and cell complexes |

81R50 | Quantum groups and related algebraic methods applied to problems in quantum theory |

18D10 | Monoidal, symmetric monoidal and braided categories (MSC2010) |

### 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
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\textit{C. Kassel}, Quantum groups. New York, NY: Springer-Verlag (1995; Zbl 0808.17003)