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The classification of simple Lie algebras over fields of positive characteristic: Methods and results. (Die Klassifikation der einfachen Lie-Algebren über Körpern mit positiver Charakteristik: Methoden und Resultate.) (German) Zbl 0782.17013
The article under review is a brief exposition of the classification problem of the simple Lie algebras over an algebraically closed field of prime characteristic \(p\). The central point of the problem is the verification of the so called generalized Kostrikin-Shafarevich conjecture that such a Lie algebra, if \(p>7\), is either classical or of generalized Cartan type, which was finally accomplished by the author and R. L. Wilson in 1991 [Bull. Am. Math. Soc., New Ser. 24, 357-362 (1991; Zbl 0725.17023)].
In this article, the main ideas and principal methods and the key steps of the proof are briefly sketched. The author estimates that there are no difficulties in principle to obtain the classification for \(p=5\) and 7, but, on the contrary, the cases \(p=2\) and 3 seem not able to be solved in the foreseeable future.
In the final section, some related topics, e.g., the classification of infinite-dimensional simple Lie algebras, support varieties, the growth of the cohomology groups, representation types, etc., are discussed.

MSC:
17B50 Modular Lie (super)algebras
17-02 Research exposition (monographs, survey articles) pertaining to nonassociative rings and algebras
17B65 Infinite-dimensional Lie (super)algebras
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