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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:
 [1] Jacobson, N.: Lie algebras. Interscience, New York (1962) · Zbl 0121.27504 [2] Benkart, GM; Isaacs, IM, On the existence of ad-nilpotent elements, Proc. Amer. Math. Soc., 63, 39-40, (1977) · Zbl 0359.17008 [3] Loday, Jean-Louis, Une version non commutative des algèbres de Lie: LES algèbres de Leibniz, Ens. Math., 39, 269-293, (1993) · Zbl 0806.55009 [4] Fialowski, A; Khudoyberdiyev, AKh; Omirov, BA, A characterization of nilpotent Leibniz algebras, Algebr. Represent. Theory, 16, 1489-1505, (2013) · Zbl 1292.17002 [5] Albeverio, SA; Ayupov, SA; Omirov, BA, Cartan subalgebras, weight spaces, and criterion of solvability of finite dimensional Leibniz algebras, Rev. Matemática Complut., 19, 183-195, (2006) · Zbl 1128.17001 [6] Ayupov, Sh.A., Omirov, B.A.: On Leibniz Algebras, Algebras and Operators Theory. In: Proceedings of the colloquium in Tashkent, pp. 1-13. Kluwer, Dordrecht (1998) · Zbl 0928.17001 [7] Varea, VR, Existence of ad-nilpotent elements and simple Lie algebras with subalgebras of codimension one, Proc. Amer. Math. Soc., 104, 363-368, (1988) · Zbl 0723.17016 [8] Cuvier, CIM, On the existence of ad-nilpotent elements, Proc. Amer. Math. Soc., 63, 39-40, (1977) · Zbl 0359.17008
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