Complete and cocomplete Lie algebras with injective and projective properties

Abstract

Motivated by the classical correspondence between short exact sequences and splitting properties in module theory, this paper examines the projective and injective analogues within the category of Lie algebras. We first show that no Lie algebra can serve as a projective or injective object with respect to arbitrary extensions, thereby clarifying the natural limitations of this analogy. To recover meaningful dual behaviors, we introduce two new notions: cocentral extensions and cocomplete Lie algebras, viewed as the natural dual counterparts of central extensions and complete Lie algebras. We prove that solvable complete Lie algebras exhibit an injective-like property, while cocomplete Lie algebras satisfying the vanishing of their second cohomology group with trivial coefficients act as projective-like objects. Moreover, we obtain a full classification of almost abelian cocomplete Lie algebras. These results establish a duality framework for completeness and cocompleteness in Lie algebra extensions, connecting structural, categorical, and cohomological aspects.