Dual Space Derivations and H2(L, F) of Modular Lie Algebras

Abstract

It is well-known that the classical vanishing results of the cohomology theory of Lie algebras depend on the characteristic of the underlying base field. The theorems of Cartan and Zassenhaus, for instance, entail that non-modular simple Lie algebras do not admit non-trivial central extensions. In contrast, early results by Block [3] prove that this conclusion loses its validity if the underlying base field has positive characteristic.Central extensions of a given Lie algebra L, or equivalently its second cohomology group H(L, F), can be conveniently described by means of derivations φ:L → L*.