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Classification of finite dimensional simple Lie algebras in prime characteristics, arxiv:math.RA/0601380 v2
, 2006
"... Abstract. We give a comprehensive survey of the theory of finite dimensional Lie algebras over an algebraically closed field of positive characteristic and announce that the classification of all finite dimensional simple Lie algebras over an algebraically closed field of characteristic p> 3 is now ..."
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Abstract. We give a comprehensive survey of the theory of finite dimensional Lie algebras over an algebraically closed field of positive characteristic and announce that the classification of all finite dimensional simple Lie algebras over an algebraically closed field of characteristic p> 3 is now complete. Any such Lie algebra is up to isomorphism either classical or a filtered Lie algebra of Cartan type or a Melikian algebra of characteristic 5. Unless otherwise specified, all Lie algebras in this survey are assumed to be finite dimensional. In the first two sections, we review some basics of modular Lie theory including absolute toral rank, generalized Winter exponentials, sandwich elements, and standard filtrations. In Section 3, we give a systematic description of all known simple Lie algebras of characteristic p> 3 with emphasis on graded and filtered Cartan type Lie algebras. We also discuss the Melikian algebras of characteristic 5 and their analogues in characteristic 3 and 2. Our main result (Theorem 7) is stated in Section 4 which also contains formulations of several important theorems frequently used in the course of classifying simple Lie algebras. The main principles of our proof of Theorem 7, with emphasis on the rank two case, are outlined in Section 5. 1. The beginnings The theory of Lie algebras over a field F of characteristic p> 0 was initiated by Jacobson, Witt and Zassenhaus. In [J 37], Jacobson has investigated purely inseparable field extensions E/F of the form E = F(c1,..., cn) where c p i ∈ F for all i ≤ n. Although such field extensions do not possess nontrivial Fautomorphisms, Jacobson has developed for them a version of Galois Theory. The rôle of Galois automorphisms in his theory was played by Fderivations. The set DerF E of all Fderivations of E carries the following three structures: • a natural structure of a vector space over E, • a natural pstructure given by the pth power map D ↦ → Dp, • a Lie algebra structure given by the commutator product. Let F denote the set of all subfields of E containing F and L the set of all Esubspaces of DerF E stable under the pth power map and Lie bracket in DerF E. Both sets F and L are partially ordered by inclusion. Given a subset X in DerF E
Simple finite group schemes and their infinitesimal deformations
, 811
"... We show that the classification of simple finite group schemes over an algebraically closed field reduces to the classification of abstract simple finite groups and of simple restricted Lie algebras in positive characteristic. Both these two simple objects have been classified. We review this classi ..."
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We show that the classification of simple finite group schemes over an algebraically closed field reduces to the classification of abstract simple finite groups and of simple restricted Lie algebras in positive characteristic. Both these two simple objects have been classified. We review this classification. Finally, we address the problem of determining the infinitesimal deformations of simple finite group schemes.