On the existence of ad-nilpotent elements
type: Item
instance of: scholarly article
title: On the existence of ad-nilpotent elements
zbMATH Open document ID: 1350.17004
author: Côme J. A. Béré
author: Marie Françoise Ouedraogo
author: Nakelgbamba Boukary Pilabré
published in: Afrika Matematika
publication date: 2015-09-23T00:00:00Z
review text: For an arbitrary element \(x\) of a Leibniz algebra \(L\) consider two operators: \[ \mathrm{ad}_x: L\rightarrow L,\quad y \rightarrow [y,x] \] and \[ \mathrm{Ad}_x:L \rightarrow L,\quad y\rightarrow [x,y]. \] Clearly, \(\mathrm{ad}_x\) and \(\mathrm{Ad}_x\) are derivation and anti-derivation of \(L\), respectively. For these operators the following relations hold true: \[ \begin{aligned} \mathrm{ad}_{[x,y]} & =\mathrm{ad}_y \mathrm{ad}_x -\mathrm{ad}_x \mathrm{ad}_y,\\ \mathrm{Ad}_{[x,y]} & =\mathrm{ad}_y \mathrm{Ad}_x - \mathrm{Ad}_x \mathrm{ad}_y,\\ \mathrm{Ad}_{[x,y]} & =\mathrm{ad}_y \mathrm{Ad}_x + \mathrm{Ad}_x \mathrm{Ad}_y,\\ 0 & =\mathrm{Ad}_x \mathrm{Ad}_y +\mathrm{Ad}_x \mathrm{ad}_y.\end{aligned} \] A (bi)module \(M\) over a Leibniz algebra \(L\) is a vector space with two (left \(l\) and right \(r\)) actions, satisfying the above relations. Let \(L\) be a Leibniz algebra and \(M\) be \(L\)-(bi)module. We denote by \(\mathrm{Ess}(M)\) the subspace of \(M\) spanned by elements of the type \(l_x(v)+r_x(v)=xv+vx\) for all \((x,v)\in L \times M.\) In the present paper the authors prove the invariance of \(\mathrm{Ess}(L)\) under derivations of \(L\) and \(\mathrm{Ess}(L)\subseteq \mathrm{Ker } \widetilde{D}\) for any anti-derivation \(\widetilde{D}\) of \(L\). Moreover, the embedding \([L_\lambda, L_\mu]\subseteq L_{\lambda+\mu}+\mathrm{Ess}(L)\), where \(L_\lambda, L_\mu\) are weight spaces with respect to a given anti-derivation, is established. The main results of the paper is the following: Theorem. Let \(L\) be a Leibniz algebra over an algebraically closed field. Let \(X\) be a non-empty subset of \(L\) such that for every \(x\in X\), all eigenvectors of \(\mathrm{ad}_x\) (correspondingly, of \(\mathrm{Ad}_x\)) lie in \(X\). Then \(\mathrm{ad}_y\) (correspondingly, \(\mathrm{Ad}_x\)) is nilpotent for some \(y\in X\).
Mathematics Subject Classification ID: 17A32
Mathematics Subject Classification ID: 17B30
zbMATH DE Number: 6484529
zbMATH Keywords: Lie algebras
zbMATH Keywords: Leibniz algebras
zbMATH Keywords: ad-nilpotent
zbMATH Keywords: derivation
zbMATH Keywords: anti-derivation
zbMATH Keywords: module
reviewed by: Sh. A. Ayupov
MaRDI profile type: Publication
full work available at URL: s13370-014-0246-y
OpenAlex ID: W2065891823
cites work: Q5732712
cites work: On the Existence of Ad-Nilpotent Elements
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cites work: Cartan subalgebras, weight spaces, and criterion of solvability of finite dimensional Leibniz algebras
cites work: Q4251228
cites work: Existence of Ad-Nilpotent Elements and Simple Lie Algebras with Subalgebras of Codimension One
DOI: 10.1007/S13370-014-0246-Y
description: scientific article; zbMATH DE number 6484529