Groups of automorphisms of Lie algebras such that the fixed-point subalgebra of each non-identity element is solvable

Abstract

Let G be a finite group of automorphisms of a non-solvable Lie algebra L of finite dimension over a field of characteristic zero. This paper is concerned with the relationship between the structure of G and that of L. By imposing the condition that the fixed-point subalgebra of each non-identity element of G is solvable, we are able to determine the structure of the group G and that of the quotient algebra of L by its solvable radical, Solv( L), except when G is a group of odd order in which every Sylow subgroup is cyclic and the center is non-trivial. In this case there exists an element x∈ L, x∉Solv( L), which is fixed by every element of G of prime order. We note that the imposed condition is satisfied by every finite group of automorphisms of a three-dimensional simple Lie algebra, regular groups of automorphisms of a non-solvable Lie algebra which are abelian of type ( p, p), p prime, and by the center of a finite group of automorphisms of a semisimple Lie algebra without proper semisimple invariant subalgebra. We exhibit several examples of groups satisfying that condition. Also, we give some results relating the structures of G and L in more general cases.