Complete and cocomplete Lie algebras with injective- and projective-type properties

Technical Summary: Complete and Cocomplete Lie Algebras with Injective- and Projective-Type Properties

Problem Statement
The paper investigates the structural properties of finite-dimensional Lie algebras over a field of characteristic zero, specifically seeking analogues to the concepts of injective and projective objects found in the category of modules. In module theory, injective and projective objects are characterized by the splitting of short exact sequences. The authors address whether similar characterizations exist in the category of Lie algebras. A fundamental complication arises because the category of Lie algebras is not abelian; consequently, extensions admit two distinct notions of splitting:

1. Semi-trivial splitting: The extension $0 \to A \to B \to C \to 0$ splits as a semidirect sum $B \cong A \rtimes_\rho C$ for some action $\rho$.
2. Trivial splitting: The extension splits as a direct sum $B \cong A \oplus_{\text{Lie}} C$.

The central questions are: Under what conditions on $A$ does every extension by $A$ split trivially? Under what conditions on $C$ does every extension of $C$ split trivially?

Methodology
The authors employ a combination of structural Lie algebra theory, homological algebra, and algorithmic computation.

• Structural Analysis: The paper utilizes Levi's theorem and Ado's theorem to analyze extension properties. It distinguishes between general extensions and central extensions (where the kernel lies in the center of the total algebra).
• Homological Characterization: The study leverages Lie algebra cohomology, specifically the second cohomology group $H^2(C, K)$, to characterize splitting properties. The authors relate the vanishing of cohomology groups to the existence of splitting homomorphisms.
• Lifting Properties: The paper reformulates splitting conditions as lifting properties for homomorphisms, paralleling the definitions of injective and projective modules but adapting them to the specific constraints of Lie algebra ideals and derivations.
• Algorithmic Implementation: The authors develop explicit algorithms based on structure constants to verify completeness and cocompleteness. These algorithms solve linear systems derived from the definitions of the center, derivations, and the coboundary operator for 2-forms.
• Classification: The methods are applied to classify complete and cocomplete Lie algebras up to dimension 4 over real and complex fields.

Key Contributions and Results

1. Injective-Type Properties and Complete Lie Algebras (Theorem 1):
   The paper establishes the converse of Jacobson's classical theorem. It proves that a Lie algebra $A$ is complete (trivial center and only inner derivations) if and only if every extension of any Lie algebra $C$ by $A$ splits trivially.

   • This identifies completeness as an "injective-type" property.
   • The authors provide four equivalent characterizations, including a homomorphism-lifting property. Crucially, they demonstrate that this lifting property requires the image of the source map to be a direct Lie summand of the target, a condition strictly weaker than full categorical injectivity (as shown in Example 3.5).
   • An algorithm (Algorithm 1) is provided to verify completeness by checking if the center is trivial and if the dimension of the derivation algebra equals the dimension of the Lie algebra.

2. Nonexistence of Global Projective Objects (Theorem 2):
   The paper proves that no nontrivial Lie algebra possesses a global "projective-type" property. Specifically, for every nontrivial Lie algebra $C$, there exists an extension by some Lie algebra $A$ that does not split trivially. This reveals a structural asymmetry between the injective and projective aspects in the category of Lie algebras.

3. Cocomplete Lie Algebras and Central Extensions (Theorem 3 & 4):
   To recover a projective analogue, the authors restrict the scope to central extensions.

   • They define a Lie algebra $C$ as cocomplete if every central extension of $C$ splits trivially.
   • Theorem 3 provides equivalent characterizations for cocompleteness, including a lifting property for homomorphisms from $C$ where the kernel of the target map is central.
   • Proposition 4.6 establishes the cohomological criterion: $C$ is cocomplete if and only if $H^2(C, K) = 0$. This implies that all semisimple Lie algebras are cocomplete.
   • Proposition 4.8 analyzes the stability of cocompleteness under direct sums, showing that a direct sum is cocomplete if and only if all summands are cocomplete and at least $m-1$ of them are perfect.

4. Almost Abelian Case (Theorem 4):
   The paper provides a necessary and sufficient spectral condition for an almost abelian Lie algebra (a 1-dimensional extension of an abelian ideal by a derivation $D$) to be cocomplete. The algebra is cocomplete if and only if $D$ is an automorphism and no pair of eigenvalues $\lambda, \mu$ of $D$ satisfies $\lambda + \mu = 0$.

5. Computational Results:
   Using the developed algorithms, the authors tabulate all complete and cocomplete Lie algebras of dimension $\le 4$ over real and complex fields. They note that while semisimple algebras are both complete and cocomplete, the class of cocomplete algebras is strictly larger (e.g., it includes the affine Lie algebra). Conversely, no non-abelian nilpotent Lie algebra is cocomplete.

Significance
The paper claims to resolve the question of injective and projective analogues in the category of Lie algebras by distinguishing between global and restricted (central) contexts.

• It clarifies that completeness is the precise structural condition for the "injective-type" behavior regarding trivial splitting of general extensions, though it is strictly weaker than categorical injectivity.
• It demonstrates that a global "projective-type" object does not exist, necessitating the introduction of cocompleteness as a dual concept restricted to central extensions.
• The work bridges structural Lie theory with homological algebra, providing explicit cohomological criteria ($H^2(C, K)=0$) and practical algorithms for identifying these classes.
• The classification of low-dimensional examples serves as a foundational dataset for further study, highlighting the interplay between semisimplicity, nilpotency, and the splitting of extensions.