A Note on Automorphisms of Lie Algebras

Abstract

In a beautiful paper which appeared in 1939 ([4]), F. Gantmacher made a thorough study of automorphisms of semi-simple Lie algebras over the field of complex numbers. Among other things, he defined the index n(G _i ) of a connected component G _i of the automorphism group G = G(8) as the minimum multiplicity of the characteristic root 1 for elements of G _i . The main purpose of this note is the determination of these indices. It is somewhat surprising that this does not appear in Gantmacher’s paper since all the methods for deriving the formula for index G _i are available in his paper. The secondary purpose of this note is to extend Gantmacher’s theory to the case of Lie algebras over arbitrary algebraically closed base fields of characteristic 0. This can be done by using algebraic group concepts and techiques which are by now well known. Nevertheless, it seems worthwhile to carry out the program in detail since Gantmacher’s results give a real insight into the action of an automorphism in a semi-simple Lie algebra. For example, as we indicate, they can be used to give a new derivation and sharpening of theorems on fixed points which are due to Borel and Mostow ([1]).