Structure theory

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The classification of finite dimensional simple Lie algebras over fields with characteristic p > 0 has been a longstanding challenge, influenced by the Kostrikin-Shafarevich Conjecture from 1966. This conjecture asserts that, for an algebraically closed field with characteristic p > 5, a finite dimensional restricted simple Lie algebra is either classical or of Cartan type. Block and Wilson proved this for p > 7 in 1988. Strade and Wilson announced a generalization for non-restricted Lie algebras in 1991, which Strade proved in 1998. The resulting Block-Wilson-Strade-Premet Classification Theorem states that every simple finite dimensional Lie algebra over an algebraically closed field of characteristic p > 3 is of classical, Cartan, or Melikian type. The author compiles the proof of this theorem in a three-volume work, aiming to provide a comprehensive account of the structure and classification of Lie algebras in positive characteristic. The first volume lays the groundwork for the classification efforts in the subsequent volumes, offering a concise overview of the general theory and classification results for a subclass of simple Lie algebras across all odd primes. This edition is corrected and serves as a valuable resource for researchers and advanced graduate students in algebra.