LEFT-SYMMETRICAL STRUCTURES ON SIMPLE MODULAR LIE-ALGEBRAS

This paper investigates left-symmetric structures on finite-dimensiona l simple Lie algebras g over a field k. If k is of characteristic 0, t hen g does not admit any left-symmetric structure. This is known in th e theory of affine manifolds. In the modular case, however, such struc tures may exist. The main purpose of this paper is to show that classi cal simple Lie algebras of characteristic p > 3 admit left-symmetric s tructures only in case p divides dim(g). The proof involves the comput ation of the first cohomology groups of classical Lie algebras for cer tain g-modules of small dimension. Here g is regarded as the Lie algeb ra of a connected semisimple algebraic group over an algebraically clo sed 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 stud ies so-called adjoint structures, induced by nonsingular derivations o f g. The simple algebra L(G, delta, f) of Block of dimension p(n) - 1, for example, admits adjoint structures for every p > 0. If p = 2, the results are more complicated.