Left-Symmetric Structures on Simple Modular Lie Algebras

Abstract

This paper investigates left-symmetric structures on finite-dimensional simple Lie algebras g over a field k. If k is of characteristic 0, then g does not admit any left-symmetric structure. This is known in the theory of affine manifolds. In the modular case, however, such structures may exist. The main purpose of this paper is to show that classical simple Lie algebras of characteristic p > 3 admit left-symmetric structures only in case p divides dim( g ). The proof involves the computation of the first cohomology groups of classical Lie algebras for certain g -modules of small dimension. Here g is regarded as the Lie algebra of a connected semisimple algebraic group over an algebraically closed field of characteristic p > 0. Most of these computations are due to Jantzen. For nonrestricted simple Lie algebras of Cartan type it is shown that many more left-symmetric structures can be found. One studies so-called adjoint structures, induced by nonsingular derivations of g . The simple algebra L ( G, δ, ƒ) of Block of dimension p n − 1, for example, admits adjoint structures for every p > 0. If p = 2, the results are more complicated.