Classification of finite dimensional simple Lie algebras in prime characteristics

We give a comprehensive survey of the theory of finite dimensional Lie algebras over an algebraically closed field of prime 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 characteristics 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. As suggested by the referee, we mention in Section 6 some interesting open problems related to the subject. We would like to thank the referee for careful reading and valuable comments.