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Here is an introduction to the theory of quantum groups with emphasis on the spectacular connections with knot theory and Drinfeld´s recent fundamental contributions. It presents the quantum groups attached to SL2 as well as the basic concepts of the theory of Hopf algebras. Coverage also focuses on Hopf algebras that produce solutions of the Yang-Baxter equation and provides an account of Drinfeld´s elegant treatment of the monodromy of the Knizhnik-Zamolodchikov equations.

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´Content.- 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 I The Hopf Algebras GL(2) and SL(2).- 5 Modules over a Hopf Algebra.- 6 Comodules.- 7 Comodule-Algebras Coaction of SL(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 Mq(2).- 4 Ring-Theoretical Properties of Mq(2).- 5 Bialgebra Structure on Mq(2).- 6 The Hopf Algebras GLq(2) and SLq(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 Afflne 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 Uq (sl(2)).- 2 Relationship with the Enveloping Algebra of sl(2).- 3 Representations of Uq.- 4 The Harish-Chandra Homomorphism and the Centre of Uq.- 5 Case when q is a Root of Unity.- 6 Exercises.- 7 Notes.- VII A Hopf Algebra Structure on Uq(sl(2)).- 1 Comultiplication.- 2 Semisimplicity.- 3 Action of Uq(sl(2))on the Quantum Plane.- 4 Duality between the Hopf Algebras Uq (sl(2))and SLq(2).- 5 Duality between Uq(sl(2))-Modules and SLq(2)-Comodules.- 6 Scalar Products on Uq(sl(2)) -Modules.- 7 Quantum Clebsch-Gordan.- 8 Exercises.- 9 Notes.- 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 GLq(2)and SLq(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 Uq(sl(2)).- 7 R-Matrices for.- 8 Exercises.- 9 Notes.- 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 Group.- 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

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