This is an introduction to the theory of affine Lie algebras and to the theory of quantum groups. It is unique in discussing these two subjects in a unified manner, which is made possible by discussing their respective applications in conformal field theory. The description of affine algebras covers the classification problem, the connection with loop algebras, and representation theory including modular properties. The necessary background from the theory of semisimple Lie algebras is also provided. The discussion of quantum groups concentrates on deformed enveloping algebras and their representation theory, but other aspects such as R-matrices and matrix quantum groups are also dealt with.
Author(s): Jürgen A. Fuchs
Series: Cambridge Monographs on Mathematical Physics
Publisher: Cambridge University Press
Year: 1992
Language: English
Pages: 447
Tags: Математика;Общая алгебра;Группы и алгебры Ли;Алгебры Ли;
Contents......Page page000010.djvu
Preface......Page page000012.djvu
1. Semisimple Lie algebras......Page page000016.djvu
1.1. Basic concepts......Page page000017.djvu
1.2. Representations and modules......Page page000026.djvu
1.3. The Killing form. Real and complex Lie algebras......Page page000034.djvu
1.4. The Cartan- Wey1 basis......Page page000038.djvu
1.5. Classification of simple Lie algebras......Page page000053.djvu
1.6. Highest weight modules......Page page000064.djvu
1.7. The Weyl group. Characters......Page page000085.djvu
1.8. Branching rules......Page page000096.djvu
1.9. Literature......Page page000102.djvu
2. Affine Lie algebras......Page page000104.djvu
2.1. Classification of Kac-Moody algebras......Page page000105.djvu
2.2. Loop algebras and central extensions......Page page000115.djvu
2.3. The root system......Page page000123.djvu
2.4. Highest weight modules......Page page000131.djvu
2.5. Null vectors......Page page000140.djvu
2.6. The Weyl group. Characters......Page page000145.djvu
2.7. Modular transformations......Page page000152.djvu
2.8. Branching rules......Page page000160.djvu
2.9. Literature......Page page000164.djvu
3. WZW theories......Page page000166.djvu
3.1. Operator products......Page page000167.djvu
3.2. The Sugawara construction......Page page000176.djvu
3.3. Conformal field theory......Page page000184.djvu
3.4. The Knizhnik-Zamo1odchikov equation......Page page000198.djvu
3.5. The Gepner-Witten equation......Page page000211.djvu
3.6. Free fermions......Page page000222.djvu
3.7. Quantum equivalence......Page page000233.djvu
3.8. Conformal embeddings......Page page000245.djvu
3.9. Literature......Page page000257.djvu
4. Quantum groups......Page page000261.djvu
4.1. Hopf algebras......Page page000262.djvu
4.2. Deformations of enveloping algebras......Page page000276.djvu
4.3. Representation theory......Page page000286.djvu
4.4. Quantum dimensions......Page page000297.djvu
4.5. The truncated Kronecker product......Page page000308.djvu
4.6. R-matrices......Page page000316.djvu
4.7. Quantized groups......Page page000324.djvu
4.8. Affine Lie algebras and quantum groups......Page page000332.djvu
4.9. Literature......Page page000338.djvu
5. Duality, fusion rules, and modular invariance......Page page000341.djvu
5.1. Fusion rule algebras......Page page000342.djvu
5.2. Modular invariance......Page page000350.djvu
5.3. Fusion rules and modular transformations......Page page000358.djvu
5.4. Modular invariants......Page page000366.djvu
5.5. WZW fusion rules and truncated tensor products......Page page000373.djvu
5.6. Chiral blocks......Page page000386.djvu
5.7. Fusing and braiding......Page page000392.djvu
5.8. Screened vertex operators and quantum groups......Page page000403.djvu
5.9. Literature......Page page000421.djvu
References......Page page000424.djvu
Index......Page page000447.djvu
