Abstract
We calculate the enveloping Lie algebras of Leibniz algebras of dimensions two and three. We show how these Lie algebras could be used to distinguish non-isomorphic (nilpotent) Leibniz algebras of low dimension in some cases. These results could be used to associate geometric objects (loop spaces) to low dimensional Leibniz algebras.
Highlights
We work with vector spaces over a field F of characteristic 0, our results can be extended in obvious way to the case of vector spaces over a field of positive characteristic, or even over a commutative ring with unit
The main tool to classify Low dimensional Leibniz algebras is to find the corresponding enveloping Lie algebra and fit them in the Beck-Kolman list of low dimensional Lie algebras. Since these Lie algebras are realized as certain quotients of the given Leibniz algebras, we first need the following fact
This theorem could be used to prove that some Leibniz algebras are non-isomorphic
Summary
We work with vector spaces (and algebras) over a field F of characteristic 0, our results can be extended in obvious way to the case of vector spaces over a field of positive characteristic (not equal 2), or even over a commutative ring with unit. By an algebra L, , we mean a vector space L over F with a (not necessarily associative) bilinear operation : L L L. We work with a class of algebras in which the left multiplication map has a stronger compatibility relation with derivations. These are Leibniz algebras, introduced by J. L. Loday [1], as non-antisymmetric generalizations of Lie algebras
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