Ebook: Quantum Groups

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.

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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.
Content:
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

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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.
Content:
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
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