| introduction 1 poisson-lie groups and lie bialgebras 1.1 poisson manifolds 1.2 poisson-lie groups 1.3 lie bialgebras 1.4 duals and doubles 1.5 dressing actions and symplectic leaves 1.6 deformation of poisson structures and quantization bibliographical notes 2 coboundary poisson-lie groups and the classical yang-baxter equation 2.1 coboundary lie bialgebras 2.2 coboundary poisson-lie groups 2.3 classical integrable systems bibliographical notes 3 solutions of the classical yang-baxter equation 3.1 constant solutions of the cybe 3.2 solutions of the cybe with spectral parameters bibliographical notes 4 quasitriangular hopf algebras 4.1 hopf algebras 4.2 quasitriangular hopf algebras bibliographical notes 5 representations and quasitensor categories 5.1 monoidal categories 5.2 quasitensor categories 5.3 invariants of ribbon tangles bibliographical notes 6 quantization of lie bialgebras 6.1 deformations of hopf algebras 6.2 quantization 6.3 quantized universal enveloping algebras 6.4 the basic example 6.5 quantum kac-moody algebras bibliographical notes 7 quantized function algebras 7.1 the basic example 7.2 r-matrix quantization 7.3 examples of quantized function algebras 7.4 differential calculus on quantum groups 7.5 integrable lattice models bibliographical notes 8 structure of que algebras:the universal r-matrix 8.1 the braid group action 8.2 the quantum weyl group 8.3 the quasitriangular structure bibliographical notes 9 specializations of que algebras 9.1 rational forms 9.2 the non-restricted specialization 9.3 the restricted specialization 9.4 automorphisms and real forms bibliographical notes 10 representations of que algebras: the generic casa 10.1 classification of finite-dimensional representations 10.2 quantum invariant theory bibliographical notes 11 representations of que algebras:the root of unity case 11.1 the non-restricted case 11.2 the restricted case 11.3 tilting modules and the fusion tensor product bibliographical notes 12 infinite-dimensional quantum groups 12.1 yangians and their representations 12.2 quantum afiine algebras 12.3 frobenius-schur duality for yangians and quantum affine algebras 12.4 yangians and infinite-dimensional classical groups 12.5 rational and trigonometric solutions of the qybe bibliographical notes 13 quantum harmonic analysis 13.1 compact quantum groups and their representations 13.2 quantum homogeneous spaces 13.3 compact matrix quantum groups 13.4 a non-compact quantum group 13.5 q-special functions bibliographical notes 14 canonical bases 14.1 crystal bases 14.2 lusztig's canonical bases bibliographical notes 15 quantum group invariants of knots and 3-manifolds 15.1 knots and 3-manifolds: a quick review 15.2 link invariants from quantum groups 15.3 modular hopf algebras and 3-manifold invariants bibliographical notes 16 quasi-hopf algebras and the knizhnik-zamolodchikov equation 16.1 quasi-hopf algebras 16.2 the kohno-drinfel'd monodromy theorem 16.3 affine lie algebras and quantum groups 16.4 quasi-hopf algebras and grothendieck's esquisse bibliographical notes appendix kac-moody algebras a 1 generalized cartan matrices a 2 kac-moody algebras a 3 the invariant bilinear form a 4 roots a 5 the weyl group a 6 root vectors a 7 aide lie algebras a 8 highest weight modules references index of notation general index |
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