Lie algebras with almost constant-free derivations
type: Item
instance of: scholarly article
title: Lie algebras with almost constant-free derivations
zbMATH Open document ID: 1162.17013
published in: Journal of Algebra
publication date: 2007-01-11T00:00:00Z
review text: 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, \(x^\delta=0\) for all \(\delta\in D\) 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 \((\mathbb Z/p\mathbb Z)\)-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).
Mathematics Subject Classification ID: 17B40
zbMATH DE Number: 5115356
zbMATH Keywords: Lie algebra
zbMATH Keywords: derivation
zbMATH Keywords: nilpotent
zbMATH Keywords: Fitting null-component
zbMATH Keywords: graded
Wikidata QID: Q115351540
author: Evgenii I. Khukhro
author: Pavel Shumyatsky
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2006.05.028
OpenAlex ID: W1992535268
cites work: Identities of graded algebras
cites work: Finite-dimensional simple Lie algebras with a nonsingular derivation
cites work: On semi-simple automorphisms of Lie algebras
cites work: On orders of finite groups and centralizers of \(p\)-elements
cites work: Finite soluble groups containing an element of prime order whose centralizer is small
cites work: Groups and Rings Having Automorphisms without Non-Trivial Fixed Elements
cites work: A Note on Automorphisms and Derivations of Lie Algebras
cites work: LIE RINGS AND LIE GROUPS ADMITTING AN ALMOST REGULAR AUTOMORPHISM OF PRIME ORDER
cites work: FINITE $ p$-GROUPS ADMITTING $ p$-AUTOMORPHISMS WITH FEW FIXED POINTS
cites work: Q4706770
cites work: Q5344241
cites work: Q5330689
cites work: Almost solubility of Lie algebras with almost regular automorphisms
cites work: Automorphisms and Fitting factors of finite groups
cites work: Automorphisms of finite groups of bounded rank
cites work: The structure of finite \(p\)-groups: Effective proof of the coclass conjectures
cites work: Pro-p groups of finite coclass
cites work: FINITE GROUPS WITH FIXED-POINT-FREE AUTOMORPHISMS OF PRIME ORDER
cites work: Automorphisms of solvable groups
DOI: 10.1016/J.JALGEBRA.2006.05.028
description: scientific article; zbMATH DE number 5115356