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-envelopesLie
algebras of special derivationsDerivation simple algebras and modulesSimple Lie
algebrasRecognition theoremsThe isomorphism problemStructure of simple Lie algebrasPairings of
induced modulesToral rank 1 Lie algebras
