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Leibniz algebras can be seen as a non-commutative generalization of Lie algebras. Thus, it is common to see a lot of papers which topic is about a generalization of a classic theorem of Lie algebras to Leibniz algebras. For instance, (1) gives a generalization of Engel's theorem to Leibniz algebras.

Is there a survey of which known theorems of Lie algebras are still valid (and also not valid) for general Leibniz algebras? If not, could we make a community wiki to gather examples? I think it would be a nice idea to put these examples in the article of Wikipedia.

(1): Ayupov, Sh A., and B. A. Omirov. "On Leibniz algebras." Algebra and operator theory. Springer, 1998. 1-12.

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    $\begingroup$ One example: if a Leibniz algebra (over any commutative ring) admit a grading in $\mathbf{Z}$ such that $\mathfrak{g}_n=0$ for all $|n|\ge n_0$ for some $n_0$ and $\mathfrak{g}_0=0$, then $\mathfrak{g}$ is nilpotent. (See my answer to mathoverflow.net/questions/90964/… ) $\endgroup$
    – YCor
    Commented Dec 29, 2014 at 16:36
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    $\begingroup$ Note that this nilpotency result is trivial for positive gradings ($\mathfrak{g}_n=0$ for all $n<0$ and all $n\ge n_0$), but the latter trivially holds for all algebras (module + bilinear law) while the former doesn't, with simple counterexamples. $\endgroup$
    – YCor
    Commented Dec 29, 2014 at 16:43

1 Answer 1

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I don't know of such a survey, but a number of important properties are established in D.W.Barnes, Some theorems on Leibniz algebras, Comm. In Alg. 39, 2463-2472, (2011) (MSN), and D. W. Barnes, On Levi’s theorem for Leibniz algebras, Bull. Aust. Math. Soc. 86, 184-185, (2012) (MSN).

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