Complete and cocomplete Lie algebras with injective and projective properties
Vu A. Le, Hoa Q. Duong, Tuan A. Nguyen
TL;DR
This paper investigates how projective and injective properties translate to Lie algebras via extension theory. It shows that no nontrivial Lie algebra is projective or injective for arbitrary extensions and introduces cocentral extensions and cocomplete Lie algebras as dual notions to central and complete Lie algebras, with a cohomological lens via $H^2$. The authors establish that complete Lie algebras have an injective-like behavior with respect to cocentral extensions, while cocomplete Lie algebras (characterized by $H^2(\\mathcal{C},\\mathbb{K}) = 0$) play a projective-like role for central extensions. They provide a full classification of almost abelian cocomplete Lie algebras by reducing the problem to proportional similarity classes of invertible matrices, and they present a structured framework linking algebraic, categorical, and cohomological aspects of Lie algebra extensions.
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.