Modular Lie Algebras (ergebnisse Der Mathematik Und Ihrer Grenzgebiete. 2. Folge)

The study of the structure of Lie algebras over arbitrary fields is now a little more than thirty years old. The first papers, to my know­ ledge, which undertook this study as an end in itself were those of JACOBSON ('Rational methods in the theory of Lie algebras') in the Annals, and of LANDHERR ('Uber einfache Liesche Ringe') in the Hamburg Abhandlungen, both in 1935. Over fields of characteristic zero, these thirty years have seen the ideas and results inherited from LIE, KILLING, E. CARTAN and WEYL developed and given new depth, meaning and elegance by many contributors. Much of this work is presented in [47, 64, 128 and 234] of the bibliography. For those who find the rationalization for the study of Lie algebras in their connections with Lie groups, satisfying counterparts to these connections have been found over general non-modular fields, with the substitution of the formal groups of BOCHNER [40] (see also DIEUDONNE [108]), or that of the algebraic linear groups of CHEVALLEY [71], for the usual Lie group. In particular, the relation with algebraic linear groups has stimulated the study of Lie algebras of linear transformations. When one admits to consideration Lie algebras over a base field of positive characteristic (such are the algebras to which the title of this monograph refers), he encounters a new and initially confusing scene. Foreword v......Page v Contents viii......Page viii 1. Definitions......Page 0001.djvu 2. The Poincare-Birkhoff-Witt theorem......Page 0005.djvu 3. Free Lie algebras. Restricted Lie algebras......Page 0006.djvu 4. Iwasawa's theorem......Page 0010.djvu 5. Nilpotent Lie algebras. Engel's theorem......Page 0011.djvu 6. Cartan subalgebras......Page 0014.djvu 7. Semisimplicity. The Killing form......Page 0015.djvu 8. Trace forms, derivations, & restrictedness......Page 0017.djvu 9. Extension of the base ring......Page 0018.djvu 1. The Cartan decomposition......Page 0021.djvu 2. Split 3-dimensional algebras & applications......Page 0024.djvu 3. Classical Lie algebras......Page 0028.djvu 4. Strings of roots & Cartan integers......Page 0030.djvu 5. Fundamental root systems......Page 0031.djvu 6. Semisimplicity and simplicity......Page 0036.djvu 7. Determination of the fundamental systems......Page 0039.djvu 8. Existence of isomorphisms......Page 0042.djvu 9. The Weyl group......Page 0044.djvu 10. Existence of the classical algebras......Page 0045.djvu 11. Generalizations of the theory......Page 0048.djvu 1. The Chevalley groups......Page 0050.djvu 2. Fundamental decomposition of G. Consequences......Page 0055.djvu 3. Structure of the Chevalley group......Page 0060.djvu 4. Conjugacy of Cartan subalgebras......Page 0063.djvu 5. Structure of the automorphism group......Page 0065.djvu 6. Realizations......Page 0066.djvu 1. Forms and splitting fields......Page 0073.djvu 2. Galois semi-automorphisms and l-cohomology......Page 0077.djvu 3. Simple involutorial algebras & the types A--D......Page 0079.djvu 4. Derivation alg. of alternative & Jordan algebras......Page 0084.djvu 5. Other types......Page 0086.djvu 6. Finite fields......Page 0088.djvu 7. On automorphism groups......Page 0093.djvu 1. Solvable and nilpotent algebras......Page 0096.djvu 2. Representations......Page 0098.djvu 3. Cohomology......Page 0101.djvu A. The Jacobson-Witt algebras Wn......Page 0105.djvu B. Some simple subalgebras of Wn......Page 0106.djvu b) The class Bn......Page 0107.djvu d) The class Tn......Page 0108.djvu b) Another generalization of the Witt algebra......Page 0109.djvu D. Isomorphisms among known simple algebras......Page 0110.djvu 5. Derivations......Page 0112.djvu 6. Extension of the base field......Page 0115.djvu 7. Cartan subalgebras......Page 0116.djvu 8. Nilpotent elements & special subalgebras......Page 0121.djvu 1. Nilpotent groups & Lie algebras. Restricted Burnside problem......Page 0126.djvu 2. Linear algebraic groups & Lie algebras......Page 0129.djvu 3. Formal groups, hyperalgebras & Lie algebras......Page 0133.djvu 4. Lie derivation alg. & purely inseparable extensions......Page 0139.djvu 5. Infinite-dim. analogues of classical Lie algebras......Page 0143.djvu Bibliography......Page 0146.djvu Index......Page 0163.djvu