x INTRODUCTION rank two, and its last application yields a crucial characterization of the Melikyan algebras, thereby completing the classification. The main goal of this monograph is to present the first complete proof of this fundamental theorem. V.G. Kac first undertook to prove the Recognition Theorem in [K2]. This pioneering work was ahead of its time. In 1970, very little was known about rational representations of simple algebraic groups in prime character- istic, and the Melikyan algebras were discovered only in the 1980s. Despite that, Kac made many deep and important observations towards the proof of the theorem in [K2]. Most of them are incorporated in Chapters 3 and 4 of this monograph. Historical accounts of the classification of simple Lie algebras of charac- teristic p 0 can be found for example in [M], [Wi3], and [B]. Investigation of the finite-dimensional simple Lie algebras over algebraically closed fields of positive characteristic began in the 1930s in the work of Jacobson, Witt, and Zassenhaus. During the next quarter century, many examples of such Lie al- gebras were discovered. In [S], written in 1967, Seligman spoke of a “rather awkward array of simple modular Lie algebras which would be totally un- expected to one acquainted only with the non-modular case.” Seligman’s book contained a characterization of the classical Lie algebras of character- istic p 3 that is, those obtained from Z-forms of the finite-dimensional simple Lie algebras over C by reducing modulo p. (See Section 2.2.) It was about the same time that Kostrikin and ˇ Safareviˇ c [KS] observed a similarity between the known nonclassical simple Lie algebras of prime characteristic and the four families W , S, H, K (Witt, special, Hamiltonian, contact) of infinite-dimensional complex Lie algebras arising in Cartan’s work on Lie pseudogroups. They called their analogues “Lie algebras of Cartan type” and formulated a conjecture which shaped research on the subject during the next thirty years. The Kostrikin-Safareviˇ ˇ c Conjecture of 1966 states Over an algebraically closed field of characteristic p 5, a finite- dimensional restricted simple Lie algebra is classical or of Car- tan type. In 1984, Block and Wilson [BW] succeeded in proving this conjecture for algebraically closed fields of characteristic p 7. If the notion of a Cartan type Lie algebra is expanded to include both the restricted and nonrestricted ones as well as their filtered deformations (determined later by Kac [K3], Wilson [Wi2], and Skryabin [Sk1]), then one can formulate the Generalized Kostrikin-Safareviˇ ˇ c Conjecture by simply erasing the word restricted in the statement above. The Generalized Kostrikin-Safareviˇ ˇ c Conjecture is now a theorem for p 7. First announced by Strade and Wilson [SW] in 1991, its proof is spread over a number of papers. As mentioned above, the proof depends in a critical way on the above Recognition Theorem (hence on our monograph).

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