Lie Algebras

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

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