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Lie Algebras
- 1st Edition, Volume 104 - July 10, 2014
- Author: Zhe-Xian Wan
- Editors: I. N. Sneddon, M. Stark
- Language: English
- Paperback ISBN:9 7 8 - 1 - 4 8 3 1 - 7 1 4 9 - 4
- eBook ISBN:9 7 8 - 1 - 4 8 3 1 - 8 7 3 0 - 3
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… Read more
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Request a sales quoteLie 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.
Preface
1. Basic Concepts
1.1. Lie Algebras
1.2. Subalgebras, Ideals and Quotient Algebras
1.3. Simple Algebras
1.4. Direct Sum
1.5. Derived Series and Descending Central Series
1.6. Killing Form
2. Nilpotent and Solvable Lie Algebras
2.1. Preliminaries
2.2. Engel's Theorem
2.3. Lie's Theorem
2.4. Nilpotent Linear Lie Algebras
3. Cartan Subalgebras
3.1. Cartan Subalgebras
3.2. Existence of Cartan Subalgebras
3.3. Preliminaries
3.4. Conjugacy of Cartan Subalgebras
4. Cartan's Criterion
4.1. Preliminaries
4.2. Cartan's Criterion for Solvable Lie Algebras
4.3. Cartan's Criterion for Semisimple Lie Algebras
5. Cartan Decompositions and Root Systems of Semisimple Lie Algebras
5.1. Cartan Decompositions of Semisimple Lie Algebras
5.2. Root Systems of Semisimple Lie Algebras
5.3. Dependence of Structure of Semisimple Lie Algebras on Root Systems
5.4. Root Systems of the Classical Lie Algebras
6. Fundamental Systems of Roots of Semisimple Lie Algebras and Weyl Groups
6.1. Fundamental Systems of Roots and Prime Roots
6.2. Fundamental Systems of Roots of the Classical Lie Algebras
6.3. Weyl Groups
6.4. Properties of Weyl Groups
7. Classification of Simple Lie Algebras
7.1. Diagrams of π Systems
7.2. Classification of Simple π Systems
7.3. The Lie Algebra G2
7.4. Classification of Simple Lie Algebras
8. Automorphisms of Semisimple Lie Algebras
8.1. The Group of Automorphisms and the Derivation Algebra of a Lie Algebra
8.2. The Group of Outer Automorphisms of a Semisimple Lie Algebra
9. Representations of Lie Algebras
9.1. Fundamental Concepts
9.2. Schur's Lemma
9.3. Representations of the Three-Dimensional Simple Lie Algebra
10. Representations of Semisimple Lie Algebras
10.1. Irreducible Representations of Semisimple Lie Algebras
10.2. Theorem of Complete Reducibility
10.3. Fundamental Representations of Semisimple Lie Algebras
10.4. Tensor Representations
10.5. Elementary Representations of Simple Lie Algebras
11. Representations of the Classical Lie Algebras
11.1. Representations of An
11.2. Representations of Cn
11.3. Representations of Bn
11.4. Representations of Dn
12. Spin Representations and the Exceptional Lie Algebras
12.1. Associative Algebras
12.2. Clifford Algebra
12.3. Spin Representations
12.4. The Exceptional Lie Algebras F4 and E8
13. Poincare-Birkhoff-Witt Theorem and Its Applications to Representation Theory of Semisimple Lie Algebras
13.1. Enveloping Algebras of Lie Algebras
13.2. Poincare-Birkhoff-Witt Theorem
13.3. Applications to Representations of Semisimple Lie Algebras
14. Characters of Irreducible Representations of Semisimple Lie Algebras
14.1. A Recursion Formula for the Multiplicity of a Weight of an Irreducible Representation
14.2. Half of the Sum of All the Positive Roots
14.3. Alternating Functions
14.4. Formula of the Character of an Irreducible Representation
15. Real Forms of Complex Semisimple Lie Algebras
15.1. Complex Extension of Real Lie Algebras and Real Forms of Complex Lie Algebras
15.2. Compact Lie Algebras
15.3. Compact Real Forms of Complex Semisimple Lie Algebras
15.4. Roots and Weights of Compact Semisimple Lie Algebras
15.5. Real Forms of Complex Semisimple Lie Algebras
Index
Other Titles in the Series in Pure and Applied Mathematics
- No. of pages: 240
- Language: English
- Edition: 1
- Volume: 104
- Published: July 10, 2014
- Imprint: Pergamon
- Paperback ISBN: 9781483171494
- eBook ISBN: 9781483187303
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