Complete and cocomplete Lie algebras with injective- and projective-type properties
In the category of modules, injective and projective objects are characterized by the splitting of short exact sequences. Motivated by this principle, we investigate analogous phenomena in the category of finite-dimensional Lie algebras over a field of characteristic zero. Since this category is not abelian, extensions admit two distinct notions of splitting, which we call \emph{trivial} and \emph{semi-trivial}. The first main result establishes the converse of Jacobson's classical theorem: a Lie algebra $\A$ trivially splits every extension by it if and only if $\A$ is complete. This identifies completeness as the ``injective-type'' property in this category -- although, seemingly weaker than category-specific injectivity. By contrast, no nontrivial Lie algebra has the dual property: the second main result asserts that, for every nontrivial Lie algebra $\Cc$, some extension of $\Cc$ fails to split trivially, so no ``projective-type'' analogue exists. Restricting to central extensions restores a workable dual notion: a Lie algebra $\Cc$ is called cocomplete if every central extension of it splits trivially, and we prove that this holds if and only if $H^2(\Cc, \K) = 0$; in particular, semisimple Lie algebras are both complete and cocomplete. Each of these results admits an equivalent homomorphism-lifting reformulation, paralleling the lifting properties of injective and projective modules. For almost abelian Lie algebras, cocompleteness reduces to an explicit spectral condition on the defining derivation. These characterizations underlie three corresponding algorithms, which yield tabulations of the complete, the cocomplete, and the almost abelian cocomplete Lie algebras of dimension at most 4.