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Modular Lie Algebras

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Table of Contents

I. Fundamentals.- 1. Definitions.- 2. The Poincare-Birkhoff-Witt theorem.- 3. Free Lie algebras. Restricted Lie algebras.- 4. Iwasawa's theorem.- 5. Nilpotent Lie algebras. Engel's theorem.- 6. Cartan subalgebras.- 7. Semisimplicity. The Killing form.- 8. Trace forms, derivations, and restrictedness.- 9. Extension of the base ring.- II. Classical Semisimple Lie Algebras.- 1. The Cartan decomposition.- 2. Split 3-dimensional algebras and applications.- 3. Classical Lie algebras.- 4. Strings of roots and Cartan integers.- 5. Fundamental root systems.- 6. Semisimplicity and simplicity.- 7. Determination of the fundamental systems.- 8. Existence of isomorphisms.- 9. The Weyl group.- 10. Existence of the classical algebras.- 11. Generalizations of the theory.- III. Automorphisms of the Classical Algebras.- 1. The Chevalley groups.- 2. The fundamental decomposition of G. Consequences.- 3. Structure of the Chevalley group.- 4. Conjugacy of Cartan subalgebras.- 5. Structure of the automorphism group.- 6. Realizations.- IV. Forms of the Classical Lie Algebras.- 1. Forms and splitting fields.- 2. Galois semi-automorphisms and 1-cohomology.- 3. Simple involutorial algebras and the types A - D.- 4. Derivation algebras of alternative and Jordan algebras.- 5. Other types.- 6. Finite fields.- 7. On automorphism groups.- V. Comparison of the Modular and Non-modular Cases.- 1. Solvable and nilpotent algebras.- 2. Representations.- 3. Cohomology.- 4. Known simple Lie algebras.- A. The Jacobson-Witt algebras Wn.- B. Some simple subalgebras of Wn.- a) The class Sn.- b) The class Vn.- c) The class Rn.- d) The class Tn.- C. Algebras defined by finite groups of functions.- a) Generalized Witt algebras.- b) Another generalization of the Witt algebra.- c) The algebras of Block.- D. Isomorphisms among known simple algebras.- 5. Derivations.- 6. Extension of the base field.- 7. Cartan subalgebras.- 8. Nilpotent elements and special subalgebras.- VI. Related Topics.- 1. Nilpotent groups and Lie algebras. The restricted Burnside problem.- 2. Linear algebraic groups and Lie algebras.- 3. Formal groups, hyperalgebras and Lie algebras.- 4. Lie derivation algebras and purely inseparable extensions.- 5. Infinite-dimensional analogues of the classical Lie algebras.

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