ON SOME STRUCTURES OF LEIBNIZ ALGEBRAS

By İsmail Demi̇r, Kailash C. Misra, E. Stitzinger. 2013

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Leibniz algebras are certain generalization of Lie algebras. In this paper we survey the important results in Leibniz algebras which are analogs of corresponding results in Lie algebras. In particular we highlight the differences between Leibniz algebras and Lie algebras. Leibniz algebras were introduced by Loday in (23) as a noncommutative gen- eralization of Lie algebras. Earlier, such algebraic structures had been considered by Bloh who called them D-algebras (11). In studying the properties of the ho- mology of Lie algebras, Loday observed that the antisymmetry of the product was not needed to prove the derived property defined on chains. This motivated him to introduce the notion of right (equivalently, left) Leibniz algebras, which is a nonas- sociative algebra with the right (equivalently, left) multiplication operator being a derivation. Thus a Leibniz algebra satisfies all defining properties of a Lie algebra except the antisymmetry of its product. Since the introduction of Leibniz algebras around 1993 several researchers have tried to find analogs of important theorems in Lie algebras. For example, it is now known that there are analogs of Lie's Theorem, Engel's Theorem, Cartan's crite- rion, and Levi's Theorem for Leibniz algebras. However, some of these results are proved for left Leibniz algebras and some are proved for right Leibniz algebras in the literature. In this paper following Barnes (7) we focus on left Leibniz algebras and state all important known results for these algebras. If some result is known only for right Leibniz algebras, then (if necessary) we include the modified proof to show that they hold for left Leibniz algebras. We include examples to show that certain results in Lie algebras do not have analogs in Leibniz algebras. Throughout this paper unless indicated otherwise all algebras and modules are assumed to be finite dimensional over an algebraically closed field F with characteristic zero.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Geometry and Topology

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