Classifying the Absolute Toral Rank Two Case

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 during the last 35 years 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 finite-dimensional simple Lie algebra over an algebraically closed field of characteristic p 3 is of classical, Cartan, or Melikian type. This is the second volume by the author, presenting the state of the art of the structure and classification of Lie algebras over fields of positive characteristic, an important topic in algebra. The contents is leading to the forefront of current research in this field.