Theorem on semisimple lie p-algebras

Abstract

The known fact on the decomposability of any semisimple Lie algebra over an algebraically closed field of characteristic zero into a direct sum of simple ideals remains true in the case of characteristic p > 0. It is also shown that a semisimple Lie p-algebra, admitting a faithful p-representation of dimension n < p−1 has such a decomposition. Its direct factors are Lie p-algebras of the classical type with a non-degenerate bilinear form of trace. The restriction n < p−1 is essential. Bibliography has 6 references.