Wan / Sneddon / Stark Lie Algebras

1. Auflage 2014
ISBN: 978-1-4831-8730-3
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band Volume 104, 240 Seiten, Web PDF

Reihe: International Series in Pure and Applied Mathematics

Lie Algebras is based on lectures given by the author at the Institute of Mathematics, Academia Sinica. This book discusses the fundamentals of the Lie algebras theory formulated by S. Lie. The author explains that Lie algebras are algebraic structures employed when one studies Lie groups. The book also explains Engel's theorem, nilpotent linear Lie algebras, as well as the existence of Cartan subalgebras and their conjugacy. The text also addresses the Cartan decompositions and root systems of semi-simple Lie algebras and the dependence of structure of semi-simple Lie algebras on root systems. The text explains in details the fundamental systems of roots of semi simple Lie algebras and Weyl groups including the properties of the latter. The book addresses the group of automorphisms and the derivation algebra of a Lie algebra and Schur's lemma. The book then shows the characters of irreducible representations of semi simple Lie algebras. This book can be useful for students in advance algebra or who have a background in linear algebra.

Autoren/Hrsg.

Wan, Zhe-Xian

Sneddon, I. N.

Stark, M.

Inhaltsverzeichnis

1;Front Cover;1
2;Lie Algebras;4
3;Copyright Page;5
4;Table of Contents;6
5;PREFACE;8
6;CHAPTER 1. BASIC CONCEPTS;10
6.1;1.1. Lie algebras;10
6.2;1.2. Subalgebras, ideals and quotient algebras;13
6.3;1.3. Simple algebras;16
6.4;1.4. Direct sum;20
6.5;1.5. Derived series and descending central series;21
6.6;1.6. Killing form;25
7;CHAPTER 2. NILPOTENT AND SOLVABLE LIE ALGEBRAS;30
7.1;2.1. Preliminaries;30
7.2;2.2. Engel's theorem;31
7.3;2.3. Lie's theorem;33
7.4;2.4. Nilpotent linear Lie algebras;35
8;CHAPTER 3. CARTAN SUBALGEBRAS;40
8.1;3.1. Cartan subalgebras;40
8.2;3.2. Existence of Cartan subalgebras;43
8.3;3.3. Preliminaries;45
8.4;3.4. Conjugacy of Cartan subalgebras;50
9;CHAPTER 4. CARTAN'S CRITERION;53
9.1;4.1. Preliminaries;53
9.2;4.2. Cartan's criterion for solvable Lie algebras;54
9.3;4.3. Cartan's criterion for semisimple Lie algebras;56
10;CHAPTER 5. CARTAN DECOMPOSITIONS AND ROOT SYSTEMS OF SEMISIMPLE LIE ALGEBRAS;57
10.1;5.1. Cartan decompositions of semisimple Lie algebras;57
10.2;5.2. Root systems of semisimple Lie algebras;62
10.3;5.3. Dependence of structure of semisimple Lie algebras on root systems;67
10.4;5.4. Root systems of the classical Lie algebras;75
11;CHAPTER 6. FUNDAMENTAL SYSTEMS OF ROOTS OF SEMISIMPLE LIE ALGEBRAS AND WEYL GROUPS;83
11.1;6.1. Fundamental systems of roots and prime roots;83
11.2;6.2. Fundamental systems of roots of the classical Lie algebras;89
11.3;6.3. Weyl groups;91
11.4;6.4. Properties of Weyl groups;95
12;CHAPTER 7. CLASSIFICATION OF SIMPLE LIE ALGEBRAS;101
12.1;7.1. Diagrams of p systems;101
12.2;7.2. Classification of simple p systems;102
12.3;7.3. The Lie algebra G2;109
12.4;7.4. Classification of simple Lie algebras;111
13;CHAPTER 8. AUTOMORPHISMS OF SEMISIMPLE LIE ALGEBRAS;114
13.1;8.1. The group of automorphisms and the derivation algebra of a Lie algebra;114
13.2;8.2. The group of outer automorphisms of a semisimple Lie algebra;117
14;CHAPTER 9. REPRESENTATIONS OF LIE ALGEBRAS;125
14.1;9.1. Fundamental concepts;125
14.2;9.2. Schur's lemma;128
14.3;9.3. Representations of the three-dimensional simple Lie algebra;129
15;CHAPTER 10. REPRESENTATIONS OF SEMISIMPLE LIE ALGEBRAS;135
15.1;10.1. Irreducible representations of semisimple Lie algebras;135
15.2;10.2. Theorem of complete reducibility;143
15.3;10.3. Fundamental representations of semisimple Lie algebras;151
15.4;10.4. Tensor representations;154
15.5;10.5. Elementary representations of simple Lie algebras;157
16;CHAPTER 11. REPRESENTATIONS OF THE CLASSICAL LIE ALGEBRAS;160
16.1;11.1. Representations of An;160
16.2;11.2. Representations of Cn;164
16.3;11.3. Representations of Bn;165
16.4;11.4. Representations of Dn;167
17;CHAPTER 12. SPIN REPRESENTATIONS AND THE EXCEPTIONAL LIE ALGEBRAS;169
17.1;12.1. Associative algebras;169
17.2;12.2. Clifford algebra;170
17.3;12.3. Spin representations;174
17.4;12.4. The exceptional Lie algebras F4 and E6;177
18;CHAPTER 13. POINCARÉ-BIRKHOFF-WITT THEOREM AND ITS APPLICATIONS TO REPRESENTATION THEORY OF SEMISIMPLE LIE ALGEBRAS;189
18.1;13.1. Enveloping algebras of Lie algebras;189
18.2;13.2. Poincaré-Birkhoff-Witt theorem;191
18.3;13.3. Applications to representations of semisimple Lie algebras;194
19;CHAPTER 14. CHARACTERS OF IRREDUCIBLE REPRESENTATIONS OF SEMISIMPLE LIE ALGEBRAS;199
19.1;14.1. A recursion formula for the multiplicity of a weight of an irreducible representation;199
19.2;14.2. Half of the sum of all the positive roots;206
19.3;14.3. Alternating functions;209
19.4;14.4. Formula of the character of an irreducible representation;212
20;CHAPTER 15. REAL FORMS OF COMPLEX SEMISIMPLE LIE ALGEBRAS;219
20.1;15.1. Complex extension of real Lie algebras and real forms of complex Lie algebras;219
20.2;15.2. Compact Lie algebras;221
20.3;15.3. Compact real forms of complex semisimple Lie algebras;224
20.4;15.4. Roots and weights of compact semisimple Lie algebras;230
20.5;15.5. Real forms of complex semisimple Lie algebras;232
21;INDEX;236
22;OTHER TITLES IN THE SERIES IN PURE AND APPLIED MATHEMATICS;238