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**The classification of the simple modular Lie algebras. I: Determination of the two-sections.**
*(English)*
Zbl 0699.17016

All known examples of finite-dimensional simple Lie algebras over an algebraically closed field of characteristic \(p>5\) are either of classical type or of generalized Cartan type. The Generalized Kostrikin- Shafarevich conjecture states that over such fields these are in fact the only finite-dimensional simple Lie algebras. This conjecture has already been verified by R. E. Block and R. L. Wilson [J. Algebra 114, 115-259 (1988; Zbl 0644.17008)] in the special case when \(p>7\) and the algebras are restricted.

In this, the first of what is intended to be a series of papers on the classification of simple modular Lie algebras, the author lays the groundwork for the project and generalizes some results which were key to the classification of the restricted simple algebras. A highlighted result is the determination of the structure of the two-sections of the simple Lie algebras.

The author’s approach to the general classification problem involves making maximum possible use of the techniques that were used and the results that were established in the process of classifying the restricted simple algebras. To do this he relies heavily on his own work on p-envelopes and absolute toral rank of a Lie algebra [Lect. Notes Math. 1373, 1-28 (1989; Zbl 0665.17006)].

In this, the first of what is intended to be a series of papers on the classification of simple modular Lie algebras, the author lays the groundwork for the project and generalizes some results which were key to the classification of the restricted simple algebras. A highlighted result is the determination of the structure of the two-sections of the simple Lie algebras.

The author’s approach to the general classification problem involves making maximum possible use of the techniques that were used and the results that were established in the process of classifying the restricted simple algebras. To do this he relies heavily on his own work on p-envelopes and absolute toral rank of a Lie algebra [Lect. Notes Math. 1373, 1-28 (1989; Zbl 0665.17006)].

Reviewer: G.Brown