Quantum Groups

Klappentext
This book provides an introduction to the theory of quantum groups with emphasis on the spectacular connections with knot theory and on Drinfeld's recent fundamental contributions. The first part presents in detail the quantum groups attached to SL2 as well as the basic concepts of the theory of Hopf algebras. Part Two focuses on Hopf algebras that produce solutions of the Yang-Baxter equation, and on Drinfeld's quantum double construction. In the following part we construct isotopy invariants of knots and links in the three-dimensional Euclidean space, using the language of tensor categories. The last part is an account of Drinfeld's elegant treatment of the monodromy of the Knizhnik-Zamolodchikov equations, culminating in the construction of Kontsevich's universal knot invariant.

Inhalt
Content.- 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 Uq(sl(2)).- Two Universal R-Matrices.- VIII The Yang-Baxter Equation and (Co)Braided Bialgebras.- IX Drinfeld's Quantum Double.- 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.- 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 the Knizhnik-Zamolodchikov Equations.- XX Postlude A Universal Knot Invariant.- References.

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