Quantum Groups

ISBN-13: 9781461269007 / Angielski / Miękka / 2012 / 534 str.

Christian Kassel

Opis

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.

Spis treści

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 Braid Group Representations.- 5 Quasi-Hopf Algebras.- 6 Exercises.- 7 Notes.- 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 Group Representations from Monodromy.- 3 The Knizhnik-Zamolodchikov Equations.- 4 The Drinfeld-Kohno Theorem.- 5 Equivalence of Uh(g)and Ag,t.- 6 Drinfeld’s Associator.- 7 Construction of the Topological Braided Quasi-Bialgebra Ag,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 Group Invariants.- 9 Exercises.- 10 Notes.- References.