Low-dimensional non-abelian Leibniz cohomology

Abstract

We construct the zero and first non-abelian cohomologies of Leibniz algebras with coefficients in crossed modules, which differ from those of Gnedbaye and generalize the zero and first Leibniz cohomologies of Loday and Pirashvili. We also introduce the second non-abelian Leibniz cohomology and describe its relationship with extensions of Leibniz algebras by crossed modules. We obtain a nine-term exact non-abelian cohomology sequence. For Lie algebras we compare the non-abelian Leibniz and Lie cohomologies.

On capability of Leibniz algebras

“…to the Lie algebra of inner derivations of a Lie algebra). To examine the similar fact for Leibniz algebras, we recall some notions from [12] (see also [5]).…”

More on crossed modules in Lie, Leibniz, associative and diassociative algebras

“…Leibniz crossed modules. First, we recall some needed notations and facts about crossed modules of Leibniz algebras from [5,12].…”

“…Let P and M be Lie algebras. Any Lie action of P on M defines in a natural way a Leibniz action of P on M , and any Lie crossed module ∂ : M → P is, at the same time, a crossed module of Leibniz algebras (see [5,Remark 11]). Thus, we have the inclusion functor XLie → XLb.…”

On non-abelian extensions of Leibniz algebras

“…Abelian extensions of Leibniz algebras is studied in [9]. Non-abelian cohomology and extensions by crossed modules of Leibniz algebras are studied in [8,16,20]. Associated to a Leibniz algebra, there is a differential graded Lie algebra structure on the graded vector space of the cochain complex, which plays an important role in studying cohomology and deformations of Leibniz algebras, see [6,14] for details.…”