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On Leibniz algebras. (English) Zbl 0928.17001
Khakimdjanov, Yusupdjan (ed.) et al., Algebra and operator theory. Proceedings of the colloquium, Tashkent, Uzbekistan, September 29–October 5, 1997. Dordrecht: Kluwer Academic Publishers. 1-12 (1998).
Leibniz algebras as introduced by Loday ten years ago are the noncommutative analogue of Lie algebras. In this paper, some of the usual Lie theory is extended to Leibniz algebras. The first section studies the notion of nilpotency. In Section 2, all three dimensional (non-Lie) Leibniz algebras are determined. In the last section it is shown that certain central Leibniz extensions of Lie algebras are again Lie algebras.
Note: Engel’s theorem for Leibniz algebras is also part of Paul Higgins’s comprehensive work [Leibniz Algebras, Oxford Thesis, 1995, 126 pages].
For the entire collection see [Zbl 0908.00015].

##### MSC:
 17A32 Leibniz algebras 17B60 Lie (super)algebras associated with other structures (associative, Jordan, etc.)