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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

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1st Edition, Volume 104 - January 1, 1975

Author: Zhe-Xian Wan

Editors: I. N. Sneddon, M. Stark

Language: EnglisheBook 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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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.

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: January 1, 1975
- Imprint: Pergamon
- eBook ISBN: 9781483187303

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