Lectures on representation theory and Knizhnik-Zamolodchikov equations / Pavel I. Etingof, Igor B. Frenkel, Alexander A. Kirillov, Jr.

Format:

Book

Author/Creator:

Etingof, P. I. (Pavel I.), 1969- author.

Frenkel, Igor, author.

Kirillov, Alexander A., 1967- author.

Series:

Mathematical surveys and monographs ; no. 58.

Mathematical surveys and monographs, 0076-5376 ; volume 58

Language:

English

Subjects (All):

Broken symmetry (Physics).

Knizhnik-Zamolodchikov equations.

Quantum groups.

Kac-Moody algebras.

Mathematical physics.

Physical Description:

1 online resource (215 p.)

Place of Publication:

Providence, Rhode Island : American Mathematical Society, [1998]

Language Note:

English

Summary:

In 13 lectures by three authors admittedly afflicted with "q-disease" (i.e. infatuation with the mathematical structures of conformal field theory and their q-deformations), they wax eruditely on this new area of theoretical physics and its powerful tools at the interface between mathematics and physics. Titles include: free field realization, quantum affine algebras, connection matrices for the quantum K-Z equations and elliptic functions, and current developments and future perspectives. Intended for those familiar with the representation theory of simple Lie algebras. Annotation copyrighted by Book News, Inc., Portland, OR

Contents:

""Contents""; ""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. Further generalizations of affine 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 equations""; ""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 differential 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=sl[sub(2)]""""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 sl[sub(2)]: 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 sl[sub(2)]""; ""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 correlation 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""

Notes:

Description based upon print version of record.

Includes bibliographical references (pages 189-195) and index.

Description based on print version record.

ISBN:

1-4704-1285-3