文摘
Basing ourselves on the categorical notions of central extensions and commutators in the framework of semi-abelian categories relative to a Birkhoff subcategory, we study central extensions of Leibniz algebras with respect to the Birkhoff subcategory of Lie algebras, called \(\mathsf {Lie}\)-central extensions. We obtain a six-term exact homology sequence associated to a \(\mathsf {Lie}\)-central extension. This sequence, together with the relative commutators, allows us to characterize several classes of \(\mathsf {Lie}\)-central extensions, such as \(\mathsf {Lie}\)-trivial extensions, \(\mathsf {Lie}\)-stem extensions and \(\mathsf {Lie}\)-stem covers, and to introduce and characterize \(\mathsf {Lie}\)-unicentral, \(\mathsf {Lie}\)-capable, \(\mathsf {Lie}\)-solvable and \(\mathsf {Lie}\)-nilpotent Leibniz algebras.
