Quantum Groups

Description

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. Front Matter....Pages i-xii Front Matter....Pages 1-1 Preliminaries....Pages 3-22 Tensor Products....Pages 23-38 The Language of Hopf Algebras....Pages 39-71 The Quantum Plane and Its Symmetries....Pages 72-92 The Lie Algebra of SL (2)....Pages 93-120 The Quantum Enveloping Algebra of sl(2)....Pages 121-139 A Hopf Algebra Structure on Uq (sl(2))....Pages 140-163 Front Matter....Pages 165-165 The Yang-Baxter Equation and (Co)Braided Bialgebras....Pages 167-198 Drinfeld’s Quantum Double....Pages 199-238 Front Matter....Pages 239-239 Knots, Links, Tangles, and Braids....Pages 241-274 Tensor Categories....Pages 275-293 The Tangle Category....Pages 294-313 Braidings....Pages 314-338 Duality in Tensor Categories....Pages 339-367 Quasi-Bialgebras....Pages 368-382 Front Matter....Pages 383-383 Generalities on Quantum Enveloping Algebras....Pages 385-402 Drinfeld and Jimbo’s Quantum Enveloping Algebras....Pages 403-419 Cohomology and Rigidity Theorems....Pages 420-448 Monodromy of the Knizhnik-Zamolodchikov Equations....Pages 449-483 Postlude. A Universal Knot Invariant....Pages 484-505 Back Matter....Pages 506-534