Quantum Groups [Paperback]

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.

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 tlÛ