Lie rings with almost regular automorphisms

Abstract

Let L be a Lie ring or a Lie algebra of arbitrary, not necessarily finite, dimension. Let φ be an automorphism of L and let CL(φ) = {a ∈ L | φ(a) = a} denote the fixed-point subring. The automorphism φ is called regular if CL(φ)= 0, that is, φ has no non-trivial fixed points. By Kreknin’s theorem [20] if a Lie ring L admits a regular automorphism φ of finite order k, that is, such that φ = 1 and CL(φ)= 0, then L is soluble of derived length bounded by a function of k, actually, by 2 − 2. (Earlier Borel and Mostow [3] proved the solubility in the finite-dimensional case, without a bound for the derived length.) In the present paper we prove that if a Lie ring admits an automorphism of prime-power order that is “almost regular,” then L is “almost soluble.”