Lie algebras with almost constant-free derivations

Abstract

Let L be a finite-dimensional Lie algebra of characteristic 0 admitting a nilpotent Lie algebra of derivations D. By a classical result of Jacobson, if D is constant-free (that is, without non-zero constants, xδ=0 for all δ∈D only if x=0), then L is nilpotent. We prove that if D is almost constant-free, then L is almost nilpotent in the following precise sense: if m is the dimension of the Fitting null-component with respect to D, then L contains a nilpotent subalgebra N of codimension bounded by a function of m and n, and the nilpotency class of N is bounded in terms of n only. This includes strengthening Jacobson's theorem by giving in the case m=0 a bound for the nilpotency class of L in terms of the number of weights of D. The main result can also be stated as a theorem about a locally finite Lie algebra over an algebraically closed field of characteristic 0 containing a nilpotent subalgebra with finitely many weights such that its Fitting null-component is of finite dimension—then the algebra contains a nilpotent subalgebra of finite codimension (with similar bounds for the codimension and the nilpotency class). The results on derivations are based on results on (Z/pZ)-graded Lie algebras, where p is a prime, with zero component of dimension m (these results in turn generalize the Higman–Kreknin–Kostrikin theorem on the nilpotency of Lie algebras with a regular automorphism of prime order).