Quantum Groups (Hardcover)

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 Quasi-Bialgebras.- 1 Quasi-Bialgebras.- 2 Braided Quasi-Bialgebras.- 3 Gauge Transformations.- 4