作者：PavelI.Etingof,I
页数：198
出版社：高等教育出版社
出版日期：2020
ISBN：9787040534986
电子书格式：pdf/epub/txt
内容简介
《表示论和Knizhnik-Zamolodchikov方程（影印版）》专门研究共形场论和q-形变中产生的数学结构。作者对Knizhnik-Zamolodchikov方程的理论和相关主题做了完整的阐述。读者不需要任何物理方面的预备知识。 该书可作为一学期研究生课程的教科书，适合对数学物理感兴趣的研究生和数学研究人员使用参考。
本书特色
本书专门研究共形场论和q-形变中产生的数学结构。作者对Knizhnik-Zamolodchikov方程的理论和相关主题做了完整的阐述。读者不需要任何物理方面的预备知识。本书可作为一学期研究生课程的教科书，适合对数学物理感兴趣的研究生和数学研究人员使用参考。
目录
Preface
Lecture 1.Introduction
1.1.Simple Lie algebras and Lie groups and their generalizations
1.2.Affine Lie algebras
1.3.Quantum groups
1.4.Knizhnik-Zamolodchikov equations
1.5.Quantum affine algebras and quantum Knizhnik-Zamolodchikov equations
1.6.Aurther generalizations of afine Lie algebras and quantum groups.
1.7.Contents of the book
Lecture 2.Representations of finite-dimensional and affine Lie algebras
2.1.Simple Lie algebras
2.2.Cartan matrices of simple Lie algebras
2.3.Highest-weight modules over simple Lie algebras and contravariant forms
2.4.Finite-dimensional representations and irreducibility of Verma modules
2.5.The maximal root ,the Coxeter numbers, and the Casimir operator
2.6.Affine Lie algebras
2.7.Verma modules and Weyl modules for affine Lie algebras
2.8.Integrable representations of affine Lie algebras
2.9.The Virasoro algebra and its action on g-modules
2.10.Generating functions and currents
Lecture 3.Knizhnik-Zamolodchikov equations
3.1.Classification of intertwining operators
3.2.Operator KZ equation
3.3.Gauge invariance of the intertwining operators
3.4.KZ equations for correlation functions
3.5.Consistency and g-invariance of the KZ equationg
3.6.Analyticity of the correlation functions
3.7.Correlation functions span the space of solutions of the KZ equations
3.8.Trigonometric form of the KZ equations
3.9.Consistent systems of diferential equations and the classical Yang’Baxter equation
Lecture 4.Solutions of the Knizhnik-Zamolodchikov equations
4.1.The simplest solution of the KZ equations for g=sl2
4.2.Simplest level one solution and Gauss hypergeometric function
4.3.Integral formulas for level one solutions
4.4.Solutions of the KZ equations for sl2:arbitrary level
4.5.Solutions of the KZ equations for a general simple Lie algebra
Lecture 5.Free field realization
5.1.Fock modules and vertex operators
5.2.Matrix elements of products of vertex operators
5.3.Interpretation of the rational part of solutions of the KZ equations
in terms of creation and annihilation operators.
5.4.Factorization of solutions of the KZ equations
5.5.Free field realization of Verma modules over sl2
5.6.Intertwining operators in the free field realization: level zero
5.7.Intertwining operators in the free field realization: positive level
5.8.Calculation of the corelation functions
Lecture 6.Quantum groups
6.1.Hopf algebras and their representations
6.2.Definition of quantum groups
6.3.Quasitriangular structure and braided tensor categories
6.4.Quantum Yang-Baxter equation and representations of braid groups
6.5.Quantum double construction
6.6.Quantum double construction for Uq (ge)
6.7.Quantum Casimir element
6.8.Intertwining operators and their commutation relations
Lecture 7.Local systems and configuration spaces
7.1.Local systems
7.2.Cohomology and homology with coeficients in local systems
7.3.Configuration spaces and Orlik-Solomon algebra
7.4.Cohomology of configuration spaces with coeficients in local systems associated with the KZ equations for sl2
7.5.Gauss-Manin connection
7.6.Relative homology
7.7.The case of arbitrary g
Lecture 8.Monodromy of Knizhnik-Zamolodchikov equations
8.1.Monodromy of KZ equations and the braid group
8.2.Asymptotics of solutions of the KZ equations
8.3.Asymptotics of the correlation functions
8.4.Monodromy with respect to an infinite base point
8.5.Commutation relations for intertwining operators
8.6.Equivalence of categories and Drinfeld-Kohno theorem
8.7.Geometric approach to equivalence of categories
Lecture 9.Quantum afine algebras
9.1.Definition of quantum affine algebras
9.2.Evaluation representations of quantum atfine algebras
9.3.Intertwining operators
9.4.Quasitriangular structure in quantum affine algebras
9.5.Factorization of the R-matrix
9.6.Evaluation representations and R-matrix for Uq (sl2)
9.7.Quantum currents
9.8.Quantum Sugawara construction in degree zero
Lecture 10.Quantum Knizhnik-Zamolodchikov equations
10.1.Operator quantum KZ equation
10.2.Quantum correlation functions
10.3.Quantum KZ equations for correlation functions
10.4.A fundamental set of solutions of the quantum KZ equations
10.5.Holonomic systems of difference equations
10.6.Analyticity of the fundamental solution of the quantum KZ equations
10.7.The noncommutative product formula for the fundamental solution
10.8.Classical limit of the quantum KZ equations
10.9.Modified quantum KZ equations
10.10 Another proof of the quantum KZ equations
Lecture 11.Solutions of the quantum Knizhnik-Zamolodchikov equations for sl2
11.1.q-analogues of classical special functions
11.2.Jackson integral
11.3.The q-hypergeometric function
11.4.Some second order difference equations
11.5.The simplest solutions of the quantum KZ equations and the q-hypergeometric function
11.6.Integral formulas for solutions
Lecture 12.Connection matrices for the quantum Knizhnik-Zamolodchikov equations and elliptic functions
12.1.Linear difference equations for functions of one complex variable
12.2.Connection relation for the q-hypergeometric equation
12.3.The connection matrix for the quantum KZ equations in the simplest case
12.4.The connection matrix and the exchange matrix for intertwining operators
Lecture 13.Current developments and future perspectives
13.1.KZ equations: quantum versus classical
13.2.Monodromy of the KZ equations, tensor categories, and quantum groups
13.3.Vertex operator algebras, conformal field theory and their g-defor-mations
13.4.Eliptic KZ equations and special values of the central charge
13.5.Double loop algebras and quantum affine algebras
13.6.Quantum KZ equations and physical models
References
Index
Lecture 1.Introduction
1.1.Simple Lie algebras and Lie groups and their generalizations
1.2.Affine Lie algebras
1.3.Quantum groups
1.4.Knizhnik-Zamolodchikov equations
1.5.Quantum affine algebras and quantum Knizhnik-Zamolodchikov equations
1.6.Aurther generalizations of afine Lie algebras and quantum groups.
1.7.Contents of the book
Lecture 2.Representations of finite-dimensional and affine Lie algebras
2.1.Simple Lie algebras
2.2.Cartan matrices of simple Lie algebras
2.3.Highest-weight modules over simple Lie algebras and contravariant forms
2.4.Finite-dimensional representations and irreducibility of Verma modules
2.5.The maximal root ,the Coxeter numbers, and the Casimir operator
2.6.Affine Lie algebras
2.7.Verma modules and Weyl modules for affine Lie algebras
2.8.Integrable representations of affine Lie algebras
2.9.The Virasoro algebra and its action on g-modules
2.10.Generating functions and currents
Lecture 3.Knizhnik-Zamolodchikov equations
3.1.Classification of intertwining operators
3.2.Operator KZ equation
3.3.Gauge invariance of the intertwining operators
3.4.KZ equations for correlation functions
3.5.Consistency and g-invariance of the KZ equationg
3.6.Analyticity of the correlation functions
3.7.Correlation functions span the space of solutions of the KZ equations
3.8.Trigonometric form of the KZ equations
3.9.Consistent systems of diferential equations and the classical Yang’Baxter equation
Lecture 4.Solutions of the Knizhnik-Zamolodchikov equations
4.1.The simplest solution of the KZ equations for g=sl2
4.2.Simplest level one solution and Gauss hypergeometric function
4.3.Integral formulas for level one solutions
4.4.Solutions of the KZ equations for sl2:arbitrary level
4.5.Solutions of the KZ equations for a general simple Lie algebra
Lecture 5.Free field realization
5.1.Fock modules and vertex operators
5.2.Matrix elements of products of vertex operators
5.3.Interpretation of the rational part of solutions of the KZ equations
in terms of creation and annihilation operators.
5.4.Factorization of solutions of the KZ equations
5.5.Free field realization of Verma modules over sl2
5.6.Intertwining operators in the free field realization: level zero
5.7.Intertwining operators in the free field realization: positive level
5.8.Calculation of the corelation functions
Lecture 6.Quantum groups
6.1.Hopf algebras and their representations
6.2.Definition of quantum groups
6.3.Quasitriangular structure and braided tensor categories
6.4.Quantum Yang-Baxter equation and representations of braid groups
6.5.Quantum double construction
6.6.Quantum double construction for Uq (ge)
6.7.Quantum Casimir element
6.8.Intertwining operators and their commutation relations
Lecture 7.Local systems and configuration spaces
7.1.Local systems
7.2.Cohomology and homology with coeficients in local systems
7.3.Configuration spaces and Orlik-Solomon algebra
7.4.Cohomology of configuration spaces with coeficients in local systems associated with the KZ equations for sl2
7.5.Gauss-Manin connection
7.6.Relative homology
7.7.The case of arbitrary g
Lecture 8.Monodromy of Knizhnik-Zamolodchikov equations
8.1.Monodromy of KZ equations and the braid group
8.2.Asymptotics of solutions of the KZ equations
8.3.Asymptotics of the correlation functions
8.4.Monodromy with respect to an infinite base point
8.5.Commutation relations for intertwining operators
8.6.Equivalence of categories and Drinfeld-Kohno theorem
8.7.Geometric approach to equivalence of categories
Lecture 9.Quantum afine algebras
9.1.Definition of quantum affine algebras
9.2.Evaluation representations of quantum atfine algebras
9.3.Intertwining operators
9.4.Quasitriangular structure in quantum affine algebras
9.5.Factorization of the R-matrix
9.6.Evaluation representations and R-matrix for Uq (sl2)
9.7.Quantum currents
9.8.Quantum Sugawara construction in degree zero
Lecture 10.Quantum Knizhnik-Zamolodchikov equations
10.1.Operator quantum KZ equation
10.2.Quantum correlation functions
10.3.Quantum KZ equations for correlation functions
10.4.A fundamental set of solutions of the quantum KZ equations
10.5.Holonomic systems of difference equations
10.6.Analyticity of the fundamental solution of the quantum KZ equations
10.7.The noncommutative product formula for the fundamental solution
10.8.Classical limit of the quantum KZ equations
10.9.Modified quantum KZ equations
10.10 Another proof of the quantum KZ equations
Lecture 11.Solutions of the quantum Knizhnik-Zamolodchikov equations for sl2
11.1.q-analogues of classical special functions
11.2.Jackson integral
11.3.The q-hypergeometric function
11.4.Some second order difference equations
11.5.The simplest solutions of the quantum KZ equations and the q-hypergeometric function
11.6.Integral formulas for solutions
Lecture 12.Connection matrices for the quantum Knizhnik-Zamolodchikov equations and elliptic functions
12.1.Linear difference equations for functions of one complex variable
12.2.Connection relation for the q-hypergeometric equation
12.3.The connection matrix for the quantum KZ equations in the simplest case
12.4.The connection matrix and the exchange matrix for intertwining operators
Lecture 13.Current developments and future perspectives
13.1.KZ equations: quantum versus classical
13.2.Monodromy of the KZ equations, tensor categories, and quantum groups
13.3.Vertex operator algebras, conformal field theory and their g-defor-mations
13.4.Eliptic KZ equations and special values of the central charge
13.5.Double loop algebras and quantum affine algebras
13.6.Quantum KZ equations and physical models
References
Index
