The problem of classifying the finite dimensional simple Lie
algebras over fields of characteristic p > 0 is a long-standing
one. Work on this question has been directed by the
Kostrikin-Shafarevich Conjecture of 1966, which states that over an
algebraically closed field of characteristic p > 5 a finite
dimensional restricted simple Lie algebra is classical or of Cartan
type. This conjecture was proved for p > 7 by Block and Wilson
in 1988. The generalization of the Kostrikin-Shafarevich Conjecture
for the general case of not necessarily restricted Lie algebras and
p > 7 was announced in 1991 by Strade and Wilson and eventually
proved by Strade in 1998. The final Block-Wilson-Strade-Premet
Classification Theorem is a landmark result of modern mathematics
and can be formulated as follows: Every simple finite dimensional
simple Lie algebra over an algebraically closed field of
characteristic p > 3 is of classical, Cartan, or Melikian type.
In the three-volume book, the author is assembling the proof of the
Classification Theorem with explanations and references. The goal
is a state-of-the-art account on the structure and classification
theory of Lie algebras over fields of positive characteristic. This
first volume is devoted to preparing the ground for the
classification work to be performed in the second and third
volumes. The concise presentation of the general theory underlying
the subject matter and the presentation of classification results
on a subclass of the simple Lie algebras for all odd primes will
make this volume an invaluable source and reference for all
research mathematicians and advanced graduate students in algebra.
The second edition is corrected. Contents Toral subalgebras in
p-envelopes Lie algebras of special derivations Derivation simple
algebras and modules Simple Lie algebras Recognition theorems The
isomorphism problem Structure of simple Lie algebras Pairings of
induced modules Toral rank 1 Lie algebras
General
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