量子群入门

Introduction

1 Poisson-Lie groups and Lie bialgebras

1.1 Poisson manifolds

A Definitions

B Functorial properties

C Symplectic leaves

1.2 Poisson-Lie groups

A Definitions

B Poisson homogeneous spaces

1.3 Lie bialgebras

A The Lie bialgebra of a Poisson-Lie group

B Martintriples

C Examples

D Derivations

1.4 Duals and doubles

A Duals of Lie bialgebras and Poisson-Lie groups

B The classical double

C Compact Poisson-Lie groups

1.5 Dressing actions and symplectic leaves

A Poisson actions

B Dressing transformations and symplectic leaves

C Symplectic leaves in compact Poisson-Lie groups

D Thetwsted ease

1.6 Deformation of Poisson structures and quantization

A Deformations of Poisson algebras

BWeylquantization

C Quantization as deformation

Bibliographical notes

2 Coboundary PoissoI-Lie groups and the classical Yang-Baxter equation

2.1 Coboundary Lie bialgebras

A Definitions

B The classical Yang-Baxter equation

C Examples

D The classical double

2.2 Coboundary Poisson-Lie groups

A The Sklyanin bracket

B r-matrices and 2-cocycles

CThe classicalR-matrix

2 3 Classical integrable systems

A Complete integrability

B Lax pairs

C Integrable systems from r-matrices

D Toda systems

Bibliographical notes

3 Solutions of the classical Yang-Baxterequation

3.1 Constant solutions of the CYBE

A The parameter space of non.skew solutions

B Description of the solutions

C Examples

D Skew solutions and quasi-Frobenins Lie algebras

3.2 Solutions of the CYBE with spectral parameters

A Clnssification ofthe solutions

B Elliptic solutions

C Trigonometrie solutions

D Rational solutions

B ibliographical notes

4 Quasitriangular Hopf algebras

4.1 Hopf algebras

A Definitions

B Examples

C Representations of Hopf algebras

D Topological Hopf algebras and duMity

E Integration Oll Hopf algebras

F Hopf-algebras

4.2 Quasitriangular Hopf algebras

A Almost cocommutative Hopf algebras

B Quasitriangular Hopf algebras

C Ribbon Hopf algebras and quantum dimension

D The quantum double

E Twisting

F Sweedler8 example

Bibliographical notes

5 Representations and quasitensor categories

5.1 Monoidal categories

A Abelian categories

B Monoidal categories

C Rigidity

D Examples

E Reconstruction theorems

5.2 Quasitensor categories

ATensorcategories

B Quasitensor categories

C Balancing

D Quasitensor categories and fusion rules

EQuasitensorcategoriesin quantumfieldtheory

5.3 Invariants of ribbon tangles

A Isotopy invariants and monoidal functors

B Tangleinvariants

CCentral ek！ments

Bibliographical notes

6 Quantization of Lie bialgebras

6.1 Deformations of Hopf algebras

A Defmitions

B Cohomologytheory

CIugiditytheorems

6.2 Quantization

A（Co-）Poisson Hopfalgebras

B Quantization

C Existence of quantizations

6.3 Quantized universal enveloping algebras

ACocommut&tiveQUE algebras

B Quasitriangular QUE algebras

CQUE duals and doubles

D The square of the antipode

6.4 The basic example

A Constmctmn of the standard quantization

B Algebra structure

C PBW basis

D Quasitriangular structure

ERepresentations

F A non-standard quantization

6.5 Quantum Kac-Moody algebras

A The-andard quantization

B The centre

C Multiparameter quantizations Bibliographical notes

7 Quantized function algebras

7.1 The basic example

A Definition

B A basis of.fn（sL2（c））

C TheR-matrixformulation

D Duality

E Representations

7.2 R-matrix quantization

A From It-matrices to bialgebras

B From bialgebras to Hopf algebras：the quantum determinant

C solutions oftheQYBE

7.3 Examples of quantized function algebras

A The general definition

B The quantum speciallinear group

C The quantum orthogonal and symplectic groups

D Multiparameter quantized function algebras

7.4 Differential calculus on quantum groups

A The de Rham complex ofthe quantum plane

BThe deRham complex ofthe quantum m×m matrices

CThedeRhamcomplex ofthe quantum generallinear group

DInvariantforms on quantumGLm

7.5 Integrable lattice models

AVertexmodels

BTransfermatrices

……

9 Specializations of QUE algebras

10 Representations of QUE algebas the generic case

11Representations of QUE algebas the root of unity case

12 Infinite-dimensionalquantum groups

13 Quantum harmonic analysis

14 Canonical bases

15 Quantum gruop invariants f knots and 3-manifolds

16 Quasi-Hopf algebras and the Knizhnik -Zamolodchikov equation

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. Drinfeld 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.