Quantum Groups

Title page
Preface
Part One Quantum SL(2)
I Preliminaries
1 Algebras and Modules
2 Free Algebras
3 The Affine Line and Plane
4 Matrix Multiplication
5 Determinants and Invertible Matrices
6 Graded and Filtered Algebras
7 Ore Extensions
8 Noetherian Rings
9 Exercises
10 Notes
II Tensor Products
1 Tensor Products of Vector Spaces
2 Tensor Products of Linear Maps
3 Duality and Traces
4 Tensor Products of Algebras
5 Tensor and Symmetric Algebras
6 Exercises
7 Notes
III The Language of Hopf Algebras
1 Coalgebras
2 Bialgebras
3 Hopf Algebras
4 Relationship with Chapter 1. The Hopf Algebras GL(2) and SL(2)
5 Modules over a Hopf Algebra
6 Comodules
7 Comodule-Algebras. Coaction of 5L(2) on the Affine Plane
8 Exercises
9 Notes
IV The Quantum Plane and Its Symmetries
1 The Quantum Plane
2 Gauss Polynomials and the q-Binomial Formula
3 The Algebra M_q(2)
4 Ring-Theoretical Properties of M_q(2)
5 Bialgebra Structure on M_q(2)
6 The Hopf Algebras GL_q(2) and SL_q(2)
7 Coaction on the Quantum Plane
8 Hopf *-Algebras
9 Exercises
10 Notes
V The Lie Algebra of SL(2)
1 Lie Algebras
2 Enveloping Algebras
3 The Lie Algebra sl(2)
4 Representations of sl(2)
5 The Clebsch-Gordan Formula
6 Module-Algebra over a Bialgebra. Action of sl(2) on the Affine Plane
7 Duality between the Hopf Algebras U(sl(2)) and SL(2)
8 Exercises
9 Notes
VI The Quantum Enveloping Algebra of sl(2)
1 The Algebra U_q(sl(2))
2 Relationship with the Enveloping Algebra of sl(2)
3 Representations of U_q
4 The Harish-Chandra Homomorphism and the Centre of U_q
5 Case when q is a Root of Unity
6 Exercises
7 Notes
VII A Hopf Algebra Structure on U_q(sl(2))
1 Comultiplication
2 Semisimplicity
3 Action of U_q(sl(2)) on the Quantum Plane
4 Duality between the Hopf Algebras U_q(sl(2)) and SL_q(2)
5 Duality between U_q(sl(2))-Modules and SL_q(2)-Comodules
6 Scalar Products on U_q(sl(2))-Modules
7 Quantum Clebsch-Gordan
8 Exercises
9 Notes
Part Two Universal R-Matrices
VIII The Yang-Baxter Equation and (Co)Braided Bialgebras
1 The Yang-Baxter Equation
2 Braided Bialgebras
3 How a Braided Bialgebra Generates R-Matrices
4 The Square of the Antipode in a Braided Hopf Algebra
5 A Dual Concept: Cobraided Bialgebras
6 The FRT Construction
7 Application to GL_q(2) and SL_q(2)
8 Exercises
9 Notes
IX Drinfeld's Quantum Double
1 Bicrossed Products of Groups
2 Bicrossed Products of Bialgebras
3 Variations on the Adjoint Representation
4 Drinfeld's Quantum Double
5 Representation- Theoretic Interpretation of the Quantum Double
6 Application to U_q(sl(2))
7 R-Matrices for U_q
8 Exercises
9 Notes
Part Three Low-Dimensional Topology and Tensor Categories
X Knots, Links, Tangles, and Braids
1 Knots and Links
2 Classification of Links up to Isotopy
3 Link Diagrams
4 The Jones-Conway Polynomial
5 Tangles
6 Braids
7 Exercises
8 Notes
9 Appendix. The Fundamental Croup
XI Tensor Categories
1 The Language of Categories and Functors
2 Tensor Categories
3 Examples of Tensor Categories
4 Tensor Functors
5 Turning Tensor Categories into Strict Ones
6 Exercises
7 Notes
XII The Tangle Category
1 Presentation of a Strict Tensor Category
2 The Category of Tangles
3 The Category of Tangle Diagrams
4 Representations of the Category of Tangles
5 Existence Proof for Jones-Conway Polynomial
6 Exercises
7 Notes
XIII Braidings
1 Braided Tensor Categories
2 The Braid Category
3 Universality of the Braid Category
4 The Centre Construction
5 A Categorical Interpretation of the Quantum Double
6 Exercises
7 Notes
XIV Duality in Tensor Categories
1 Representing Morphisms in a Tensor Category
2 Duality
3 Ribbon Categories
4 Quantum Trace and Dimension
5 Examples of Ribbon Categories
6 Ribbon Algebras
7 Exercises
8 Notes
XV Quasi-Bialgebras
1 Quasi-Bialgebras
2 Braided Quasi-Bialgebras
3 Gauge Transformations
4 Braid Group Representations
5 Quasi-Hopf Algebras
6 Exercises
7 Notes
Part Four Quantum Groups and Monodromy
XVI Generalities on Quantum Enveloping Algebras
1 The Ring of Formal Series and h-Adic Topology
2 Topologically Free Modules
3 Topological Tensor Product
4 Topological Algebras
5 Quantum Enveloping Algebras
6 Symmetrizing the Universal R-Matrix
7 Exercises
8 Notes
9 Appendix. Inverse Limits
XVII Drinfeld and Jimbo's Quantum Enveloping Algebras
1 Semisimple Lie Algebras
2 Drinfeld-Jimbo Algebras
3 Quantum Group Invariants of Links
4 The Case of sl(2)
5 Exercises
6 Notes
XVIII Cohomology and Rigidity Theorems
1 Cohomology of Lie Algebras
2 Rigidity for Lie Algebras
3 Vanishing Results for Semisimple Lie Algebras
4 Application to Drinfeld-Jimbo Quantum Enveloping Algebras
5 Cohomology of Coalgebras
6 Action of a Semisimple Lie Algebra on the Cobar Complex
7 Computations for Symmetric Coalgebras
8 Uniqueness Theorem for Quantum Enveloping Algebras
9 Exercises
10 Notes
11 Appendix. Complexes and Resolutions
XIX Monodromy of the Knizhnik-Zamolodchikov Equations
1 Connections
2 Braid Croup Representations from Monodromy
3 The Knizhnik-Zamolodchikov Equations
4 The Drinfeld-Kohno Theorem
5 Equivalence of U_h(g) and A_{g,t}
6 Drinfeld's Associator
7 Construction of the Topological Braided Quasi-Bialgebra A_{g,t}
8 Verification of the Axioms
9 Exercises
10 Notes
11 Appendix. Iterated Integrals
XX Postlude. A Universal Knot Invariant
1 Knot Invariants of Finite Type
2 Chord Diagrams and Kontsevich's Theorem
3 Algebra Structures on Chord Diagrams
4 Infinitesimal Symmetric Categories
5 A Universal Category for Infinitesimal Braidings
6 Formal Integration of Infinitesimal Symmetric Categories
7 Construction of Kontsevich's Universal Invariant
8 Recovering Quantum Croup Invariants
9 Exercises
10 Notes
References
Index