Lie algebras with almost constant-free derivations

Khukhro, E. I. and Shumyatsky, P. (2006) Lie algebras with almost constant-free derivations. Journal of Algebra, 306 (2). pp. 544-551. ISSN 0021-8693

Full content URL: http://dx.doi.org/10.1016/j.jalgebra.2006.05.028

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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 / p Z)-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). © 2006 Elsevier Inc. All rights reserved.

Keywords:Algebra, Groups
Subjects:G Mathematical and Computer Sciences > G100 Mathematics
Divisions:College of Science > School of Mathematics and Physics
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ID Code:15654
Deposited On:14 Nov 2014 13:22

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