Lie algebras with almost constant-free derivations

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• 作者：E.I. Khukhro ; P. Shumyatsky
• 关键词：Lie algebra ; Derivation ; Nilpotent ; Fitting null-component ; Graded
• 刊名：Journal of Algebra
• 出版年：2006
• 出版时间：15 December 2006
• 年：2006
• 卷：306
• 期：2
• 页码：544-551
• 全文大小：124 K

文摘

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, x^δ=0 for all δD 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 -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).