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Quantum groups and knot invariants. (English) Zbl 0878.17013
Panoramas et Synthèses. 5. Paris: Société Mathématique de France. vi, 115 p. (1997).
This survey is an introduction to quantum groups, braided categories, knots, three-manifolds and their invariants. The first two algebraic topics are intended as preparation for the last three topological topics. Of the book’s nine chapters, the first four represent the algebraic side and the last five the topological side. Although it is concisely written, proofs are given. While only certain aspects of the algebraic aspects of quantum groups are given, namely those needed for the topological applications, the book could be used as a text for a course which intends to do the topological topics. There are examples and exercises, and some proofs would have to have details filled in by the instructor and students. For this purpose, and for a general reference on its topics, it seems an excellent source.
We indicate some of the topics. After discussing the Yang-Baxter equation, braided monoidal categories are introduced, the main example being modules over a braided bialgebra. Then chapters on the Drinfeld double and quantum universal enveloping algebras of the special linear Lie algebras complete the algebraic part. The topological part starts with the Jones polynomial, showing how knots, links and tangles lead to braided monoidal categories. Certain such categories called ribbon categories are introduced, examples coming from representations of quantum groups and from tangles. In this connection, isotopy invariants of knots, links and tangles are constructed, eventually leading to the Reshetikhin-Turaev invariant of three-manifolds. One chapter is devoted to Vassiliev invariants of links. The last chapter presents more advanced topics based on Drinfeld’s work, in particular Kontsevich’s construction of the universal link invariant and its use in recovering quantum invariants.
Once again, this book is highly recommended for mathematicians of all levels wishing to learn about these topics, as a reference book and as a textbook for a certain type of graduate course.

MSC:
17B37 Quantum groups (quantized enveloping algebras) and related deformations
57M25 Knots and links in the \(3\)-sphere (MSC2010)
17-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to nonassociative rings and algebras
57-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to manifolds and cell complexes
18D10 Monoidal, symmetric monoidal and braided categories (MSC2010)
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