Universal Central Extensions in Semi-Abelian Categories

Abstract

Abstract. Basing ourselves on Janelidze and Kelly's general notion of central extension, we study universal central extensions in the context of semi-abelian categories. We consider a new fundamental condition on composition of central extensions and give examples of categories which do, or do not, satisfy this condition.

A study of n-Lie-isoclinic Leibniz algebras

“…Let m, n be two-sided ideals of a Leibniz algebra g. The following notions come from [7], which were derived from [8].…”

“…This philosophy comes from the categorical theory of central extensions relative to a chosen Birkhoff subcategory of a semiabelian category. We refer to [3,7,8] and references given therein for a detailed explanation.…”

The c-nilpotent Schur Lie-multiplier of Leibniz algebras

“…Leibniz algebras constitute a variety of Ω-groups [18], hence it is a semi-abelian variety [14,21] denoted by Leib, whose morphisms are linear maps that preserve the Leibniz bracket.…”

“…The general theory of central extensions relative to a chosen subcategory of a base category was introduced in [20], where the simultaneous categorical and Galois theoretic approach is based on, and generalizes, the work of the Fröhlich school [16,17,25] which focused in varieties of Ω-groups, was considered in the context of semi-abelian categories [21] relative to a Birkhoff subcategory in [14,15]. Examples like groups vs. abelian groups, Lie algebras vs. vector spaces are absolute, meaning that they fit in the general theory when the considered Birkhoff subcategory is the subcategory of all abelian objects.…”

A Non-abelian Tensor Product of Hom–Lie Algebras

“…there exists one and only one homomorphism of Hom-Lie algebras h : The category HomLie is an example of a semi-abelian category which does not satisfy universal central extension condition in the sense of [4], that is, the composition of central extensions of Hom-Lie algebras is not central in general, but it is an α-central extension (see Theorem 4.3(a) below). This fact does not allow complete generalization of classical results to Hom-Lie algebras and the well-known properties of universal central extensions are divided between universal central and universal α-central extensions of Hom-Lie algebras.…”