Lectures on Quantum Groups

Transcription

1 Lectures in Mathematical Physics Lectures on Quantum Groups Pavel Etingof and Olivier Schiffinann Second Edition International Press * s. c *''.. \ir.ik,!.'.....

2 Contents Introduction ix 1 Poisson algebras and quantization Modules over rings of power series Topologically free if-modules Completion of if-modules Poisson algebras Definition Examples of Poisson algebras Quantization of Poisson algebras Deformations Quantization Examples of quantization Loss of symmetry in quantization Poisson manifolds and quantization Definition Symplectic leaves of a Poisson manifold Quantization of Poisson manifolds Example of quantization of a Poisson manifold (Geometric quantization) ' Rational forms of a quantization Physical meaning of quantization Poisson-Lie groups Poisson-Lie groups Definition Lie bialgebras Definition Examples of Lie bialgebras Duality Poisson-Lie theory Main theorem of Poisson-Lie theory Dual Poisson-Lie group Examples of dual Lie bialgebras and dual Poisson-Lie groups 22 3 Coboundary Lie bialgebras Some Lie algebra cohomology Coboundary Lie bialgebras The classical Yang-Baxter map Triangular Lie bialgebras and the classical Yang-Baxter equation 28 iii

3 iv CONTENTS 3.5 Classification of triangular structures Quasitriangular Lie bialgebras Examples of coboundary, triangular and quasitriangular Lie bialgebras 32 4 Drinfeld's double construction Manin triples Drinfeld's double Examples Standard Lie bialgebra structure on simple Lie algebras Notations Standard structure 39 5 Belavin-Drinfeld classification (I) Coboundary structure on simple Lie bialgebras Skew-symmetric r-matrices Non skew-symmetric r-matrices Proof of the classification theorem The Cayley transform Proof of part 1) Proof of part 2) 47 6 Infinite dimensional Lie bialgebras Infinite Manin triples Examples The standard structure on Kac-Moody algebras The CYBE with spectral parameter \ An example: the Yangian and its dual The CYBE with spectral parameter Construction of a Lie bialgebra from an r-matrix Solutions of the CYBE with spectral parameters Affine Lie algebras Definition Lie bialgebra structure 59 7 Belavin-Drinfeld classification (II) Properties of nondegenerate solutions Meromorphic continuation of r{z) to C Proof of the classification theorem Myberg's theorem Elliptic solutions Rational and trigonometric r-matrices 65

4 CONTENTS v 8 Hopf algebras Definition of Hopf algebras Finite groups revisited 68~ Coalgebras Hopf algebras Pictorial representation Examples of Hopf algebras Duality in Hopf algebras Deformation Hopf algebras 76 9 Quantized universal enveloping algebras Quantized enveloping algebras The quantization theorem Examples Coboundary, quasitriangular, triangular Hopf algebras Coboundary Hopf algebras (Quasi)triangular Hopf algebras Modifications of the quantization theorem Quantization by twists Formal groups and /i-formal groups Definition Duality R-matrices and R-forms Comodules Universal R-forms (coquasitriangular structures) Infinite dimensional quantum groups The RTT formalism and h-formal groups Formal groups revisited The RTT formalism Examples RTT formalism and quantum groups Examples TheYangian The dual Yangian Quantum elliptic algebra Quantized affine Lie algebra The quantum double The quantum double The quantum double for quantized universal enveloping algebras Quasitriangular structure on Uh(g) HI

5 vi CONTENTS 13 Tensor categories and quasi-hopf algebras Semigroup categories Definition Examples Tensor functors Monoidal categories Units in semigroup categories MacLane's theorem Quasi-bialgebras and quasi-hopf algebras Definition Equivalence of quasi-bialgebras and twists "Nonabelian cohomology" Braided tensor categories Braided monoidal categories Motivation The braid group Braided tensor functors Braid group representations Symmetric categories Quasitriangular Quasi-Hopf algebras Equivalence of quasitriangular quasi-hopf algebras KZ equations and the Drinfeld Category The KZ equations: :1 Definition "Link with the CYBE Monodromy of the KZ equations, The KZ associator L Quasi-Hopf structure Braided (quasitriangular) structure The Drinfeld category Braid group representation Quasi-Hopf quantized enveloping algebras Quasi-Hopf quantized enveloping algebras Definition Examples Twists Lie quasibialgebras Definition Quantization of Lie quasibialgebras Quasitriangular Lie quasibialgebras Twists Associators Definition 150

6 CONTENTS vii Action of twists on Ass(g, ft) Classification of quasitriangular quasi-hopf QUE algebras The Drinfeld-Kohno theorem Geometric interpretation of Lie quasibialgebras Lie associators Lie associators Definition The space of Lie associators The Grothendieck-Teichmuller group Definition The action of GTi(k) on completed braid groups Drinfeld's conjecture Fiber functors and Tannaka-Krein duality Tensor categories Fiber functor First example Tannaka-Krein duality Tannaka-Krein duality for bialgebras Quantization of finite dimensional Lie bialgebras, Quantization of the Drinfeld double The Drinfeld category The forgetful functor The Verma modules Tensor structure on the forgetful functor Quantization of g Quantization of finite-dimensional Lie bialgebras Quasitriangular quantization Quantization of r-matrices Universal constructions Cyclic categories Definition Basic notions related to cyclic categories Linear algebraic structures Universal constructions Acyclic tensor calculus Universal quantization Statement of the theorem Quantization of finite-dimensional Lie bialgebras revisited Categorical Drinfeld double The Drinfeld category Quantization of g+ 201

7 viii CONTENTS 21.6 Quantization of Poisson-Lie groups DEQUANTIZATION AND THE EQUIVALENCE THE QUANTUM DOUBLE IN A SYMMETRIC Dequantization KZ ASSOCIATOR AND MULTIPLE The multiple zeta function Multiple zeta values and the KZ equation The relations between multiple zeta values Solutions to Problems and Exercises 214