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On the existence of ad-nilpotent elements. (English) Zbl 1350.17004
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 \(y\in X\).
MSC:
17A32 Leibniz algebras
17B30 Solvable, nilpotent (super)algebras
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References:
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