Simple Lie Algebras over Fields of Positive Characteristic: I. Structure Theory

The first volume of a two-volume monograph contains a complete review of methods and results of classification of simple Lie algebras over an algebraically closed field of characteristic p>3. A substantial classification scheme in the remaining characteristics p=2, 3, is also presented. Indispensable tools for the classification scheme in this case, mainly a p-envelope and absolute toral rank, special derivations and their Witt algebra, are discussed in the first three chapters. In the fourth chapter, simple algebras of classical, Cartan and Melikian type are introduced. It is shown that the latter two algebras carry a distinguished natural filtration. Here the reader can also find a list of all presently known simple Lie algebras in characteristic 3. The fifth chapter contains, employing cohomology theory, various recognition theorems based on an observation that a graded Lie algebra is determined by its non-positive part. In the remaining chapters, a complete solution of the isomorphism problem of classical, Cartan and Melikian algebras is given. Also, derivation algebras and automorphism groups of the latter two algebras are determined.

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