Here the authors give a comprehensive survey of the theory of finite dimensional Lie algebras over an algebraically closed field of characteristic \(p>0\) and announce that for \(p>3\) the classification of finite dimensional simple Lie algebras is complete (the proof has been given in their paper in J. Algebra 314, No. 2, 664–692 (2007; Zbl 1139.17007)). Any such Lie algebra is up to isomorphism either classical (i.e. comes from characteristic 0) or a filtered Lie algebra of Cartan type or a Melikian algebra of characteristic 5. A list of open problems has been added as suggested by the referee, and a comprehensive list of references is given.
The interested reader should also read the second author’s monograph “Simple Lie algebras over fields of positive characteristic. I: Structure theory. de Gruyter Expositions in Mathematics 38. Berlin: de Gruyter (2004; Zbl 1074.17005) for a profound exposition of the theory.
For the entire collection see [Zbl 1097.20500].