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 characteristic p>0 and announce that for p>3 the classification of finite dimensional simple Lie algebras is complete. 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.Comment: Revised version: a list of open problems has been added as suggested by the refere

Simple Lie algebras of small characteristic VI. Completion of the classification

Let L be a finite-dimensional simple Lie algebra over an algebraically closed field of F characteristic p>3. We prove that if the p-envelope of L in the derivation algebra of L contains nonstandard tori of maximal dimension, then p=5 and L is isomorphic to one of the Melikian algebras. Together with our earlier results this implies that any finite-dimensional simple Lie algebra over F is of classical, Cartan or Melikian type.Comment: Many typos corrected and introduction extended; the new version is accepted for publicatio

The classification of the simple modular Lie algebras II. The toral structure

AbstractLet L be a simple Lie algebra over an algebraically closed field of characteristic p > 7 and T an optimal torus in some p-envelope Lp. We determine the action of T on the two-sections of L, which have been described in [St4]. We also give some new and noncomputational proofs to determine the conjugacy classes of the tori in W(n; 1) and of the Cartan subalgebras of W(1; n)

Simple Lie algebras of small characteristic IV. Solvable and classical roots

AbstractLet L be a finite-dimensional simple Lie algebra over an algebraically closed field F of characteristic p>3 and T a torus of maximal dimension in the p-envelope of L in DerL. In this paper we describe the T-semisimple quotients of the 2-sections of L relative to T and prove that if all 1-sections of L relative to T are compositionally classical or solvable then L is either classical or a Block algebra or a filtered Lie algebra of type S

Commutative 2-cocycles on Lie algebras

On Lie algebras, we study commutative 2-cocycles, i.e., symmetric bilinear forms satisfying the usual cocycle equation. We note their relationship with antiderivations and compute them for some classes of Lie algebras, including finite-dimensional semisimple, current and Kac-Moody algebras.Comment: v7: minor changes; added ancillary file with GAP cod

Block algebras with HH1 a simple Lie algebra

The purpose of this note is to add to the evidence that the algebra structure of a p-block of a finite group is closely related to the Lie algebra structure of its first Hochschild cohomology group. We show that if B is a block of a finite group algebra kG over an algebraically closed field k of prime characteristic p such that HH1(B) is a simple Lie algebra and such that B has a unique isomorphism class of simple modules, then B is nilpotent with an elementary abelian defect group P of order at least 3, and HH1(B) is in that case isomorphic to the Witt algebra HH1(kP)⁠. In particular, no other simple modular Lie algebras arise as HH1(B) of a block B with a single isomorphism class of simple modules

Lie solvable enveloping algebras of characteristic two

Lie solvable restricted enveloping algebras were characterized by Riley and Shalev except when the ground field is of characteristic 2. We resolve the characteristic 2 case here which completes the classification. As an application of our result, we obtain a characterization of ordinary Lie algebras over any field whose enveloping algebra is Lie solvable