Homology and central extensions of Leibniz and Lie $n$-algebras

Abstract

From the viewpoint of semi-abelian homology, some recent results on homology of Leibniz n-algebras can be explained categorically. In parallel with these results, we develop an analogous theory for Lie n-algebras. We also consider the relative case: homology of Leibniz n-algebras relative to the subvariety of Lie n-algebras.

Some aspects of semi-abelian homology and protoadditive functors

“…For example, this has been done in [15] for A the variety of precrossed modules and B its subvariety of crossed modules, or for A the variety of groups and B its subvariety of nilpotent groups of a fixed class k ≥ 1 (see [8]), or the variety of solvable groups of a fixed class k ≥ 1. Similar results have been obtained in the categories of Leibniz and of Lie n-algebras in [7].…”

Higher central extensions and cohomology

“…for all I Ď n. The category X satisfies the commutator condition on ncubic central extensions when the H-central n-cubic extensions in X coincide with the categorically central ones, namely those which are central with respect to AbpX q in the Galois-theoretic sense used throughout the rest of the paper. We say that X satisfies the commutator condition (CC) when it satisfies the commutator condition on n-cubic central extensions for all n. [30]. Moreover, from [79] we know that any semi-abelian category with the Smith is Huq condition has (CC), while the categories of loops and of commutative loops do not satisfy this condition.…”

“…It is shown in [40] that, next to the category of groups, also the categories Lie algebras and non-unitary rings have (CC). The examples of Leibniz and Lie n-algebras were treated in [30]. Moreover, from [79] we know that any semi-abelian category with the Smith is Huq condition has (CC), while the categories of loops and of commutative loops do not satisfy this condition.…”

“…Of course, whether or not an n-cubic extension is central with respect to some chosen Birkhoff subcategory depends on this subcategory more than anything else. In many cases (like, for instance, the case of groups vs. 2-nilpotent groups) there are explicit descriptions of the central extensions in some, or in all, degrees (see, for instance, [30,39,38,42]). Knowing, in a given case, what the central extensions are, gives a complete description of the corresponding homology objects as higher Hopf formulae: this is the content of Theorem 8.1 in [40].…”