量子群入門

內(nèi)容概要

quantum groups first arose in the physics literature, particularly in the work of L. D. Faddeev and the Leningrad school, from the 'inverse scattering method', which had been developed to construct and solve 'integrable' quantum systems. They have excited great interest in the past few years because of their unexpected connections with such, at first sight, unrelated parts of mathematics as the construction of knot invariants and the representation theory of algebraic groups in characteristic p.　　In their original form, quantum groups are associative algebras whose defin-ing relations are expressed in terms of a matrix of constants (depending on the integrable system under consideration) called a quantum R-matrix. It was realized independently by V. G. Drinfel'd and M. Jimbo around 1985 that these algebras are Hopf algebras, which, in many cases, are deformations of 'universal enveloping algebras' of Lie algebras. A little later, Yu. I. Manin and S. L. Woronowicz independently constructed non-commutative deforma-tions of the algebra of functions on the groups SL2(C) and SU2, respectively,and showed that many of the classical results about algebraic and topological groups admit analogues in the non-commutative case.

書籍目錄

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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