Helmut Strade Knihy

The classification of finite dimensional simple Lie algebras over fields of 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 Block-Wilson-Strade-Premet Classification Theorem is a significant milestone, stating that every simple finite dimensional Lie algebra over an algebraically closed field of characteristic p > 3 is of classical, Cartan, or Melikian type. This volume is the second of a three-volume series on classifying simple Lie algebras over algebraically closed fields of characteristic > 3. The first volume covers methods and initial classification results, while this second volume delves into the structure of tori in Hamiltonian and Melikian algebras. It provides a complete proof for classifying absolute toral rank 2 simple Lie algebras, utilizing sandwich element methods and insights from filtered and graded Lie algebras.

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.

This is the last of three volumes about "Simple Lie Algebras over Fields of Positive Characteristic"by Helmut Strade, presenting the state of the art of the structure and classification of Lie algebras over fields of positive characteristic.