Almost Abelian Cocomplete Lie Algebras

• Almost abelian cocomplete Lie algebras are finite-dimensional structures with a codimension-one abelian ideal and vanishing H² that ensures every central extension splits.
• Their classification hinges on the invertibility of the derivation D and a spectral condition that prohibits any eigenpair from summing to zero.
• This framework highlights a duality in Lie algebra theory, linking cocompleteness with extension splitting and contrasting with the concept of complete Lie algebras.

An almost abelian cocomplete Lie algebra is a finite-dimensional Lie algebra over a field of characteristic zero that admits a codimension-one abelian ideal and possesses the splitting property for all central extensions by abelian Lie algebras, equivalent to the vanishing of its second Lie algebra cohomology group with trivial coefficients. The complete classification of such structures exposes deep cohomological constraints on the allowable derivations and eigenstructures, distinguishing these algebras as the natural duals to complete Lie algebras in the categorical theory of extensions and splitting.

1. Definition and Structural Characterization

Let $\mathfrak{g}$ be a finite-dimensional Lie algebra over a field $\mathbb{F}$ of characteristic zero. $\mathfrak{g}$ is almost abelian if it contains a codimension-one abelian ideal. Equivalently, there is a decomposition

$\mathfrak{g} = \mathbb{F}e \oplus V,$

where $V$ is abelian ($[v, w] = 0$ for all $v, w \in V$) and $e \notin V$, with nonzero brackets of the form $[e, v] = A(v)$ for all $v \in V$, where $A: V \to V$ is a linear endomorphism. The Lie structure is completely determined by $A$, which is necessarily a derivation of the abelian algebra $V$ (Le et al., 16 Nov 2025).

2. Cocomplete Lie Algebras and Central Extensions

For any Lie algebra $\mathfrak{g}$, a central extension by an abelian vector space $V$ is a short exact sequence

$$0 \rightarrow V \rightarrow \mathfrak{b} \rightarrow \mathfrak{g} \rightarrow 0$$

with $V$ in the center of $\mathfrak{b}$. Such an extension splits trivially if $\mathfrak{b} \cong V \oplus \mathfrak{g}$ as Lie algebras. A Lie algebra is cocomplete if every central extension by an abelian $V$ splits trivially. This condition is equivalent to the vanishing of the second cohomology group with trivial coefficients:

$H^2(\mathfrak{g}, \mathbb{F}) = 0.$

Equivalence classes of central extensions are parametrized by $H^2(\mathfrak{g}, V)$, and since $V$ is a trivial module, $$H^2(\mathfrak{g}, V) \cong H^2(\mathfrak{g}, \mathbb{F}) \otimes V$$; thus, all central extensions split if and only if $H^2(\mathfrak{g}, \mathbb{F}) = 0$ (Le et al., 16 Nov 2025).

3. Classification of Almost Abelian Cocomplete Lie Algebras

Consider $\mathfrak{g} = V^n \oplus_D \mathbb{F}e_0$ with $[e_0, v] = D(v)$ and $V^n$ abelian. The complete cocomplete property for almost abelian Lie algebras is characterized by the following theorem:

Theorem: For such an algebra, the following are equivalent:

• (i) $\mathfrak{g}$ is cocomplete ($H^2(\mathfrak{g}, \mathbb{F}) = 0$).
• (ii) $D \in \mathrm{GL}(V^n)$ is invertible, and for the eigenvalues $\lambda, \mu$ of $D$ (over $\mathbb{F}$ or its complexification), $\lambda + \mu \neq 0$ for all pairs.

Thus, the derivation $D$ must be non-singular, and no sum of two eigenvalues can be zero. This excludes, for example, any $D$ with an eigenpair $\{\alpha, -\alpha\}$ (Le et al., 16 Nov 2025).

<table> <thead> <tr> <th>Condition</th> <th>Interpretation</th> <th>Consequence</th> </tr> </thead> <tbody> <tr> <td>$D$ invertible</td> <td>$\det(D) \neq 0$</td> <td>No nontrivial center</td> </tr> <tr> <td>No eigenpairs $\lambda + \mu = 0$</td> <td>Forbids symmetric spectra</td> <td>Precludes certain decompositions</td> </tr> </tbody> </table>

4. Cohomological Structure: Hochschild–Serre Computation

The explicit computation of $H^2(\mathfrak{g}, \mathbb{F})$ leverages the Hochschild–Serre spectral sequence for the codimension-one abelian ideal $V$:

$$H^2(\mathfrak{g}, \mathbb{F}) \cong \Lambda^2_D(V^*) \oplus V^*/\mathrm{Im}(D^\top)$$

where:

• $V^*/\mathrm{Im}(D^\top) = 0$ if and only if $D^\top$ is surjective, i.e., $D$ invertible.
• $\Lambda^2_D(V^*)$ is the subspace of two-forms $\omega$ such that $$(D^\top\omega)(v_1, v_2) = \omega(Dv_1, v_2) + \omega(v_1, Dv_2) = 0$$ for all $v_1, v_2 \in V$. Nonzero $\omega$ of the type $v^* \wedge w^*$ exist in $\Lambda^2_D(V^*)$ precisely when $D$ admits two eigenvalues summing to zero.

Hence, cocompleteness is equivalent to the condition that $D$ is invertible and has no eigenpair with vanishing sum (Le et al., 16 Nov 2025).

5. Explicit Low-Dimensional Cases

For dimensions $2$ and $3$, the classification yields concrete normal forms:

Dimension 2 ($n=1$): $V = \mathbb{F} v$, $D(v) = \lambda v$, algebra defined by $[e, v] = \lambda v$.

• $D$ invertible $\Longleftrightarrow$ $\lambda \neq 0$.
• No sum-zero pair $\Longleftrightarrow$ $2\lambda \neq 0$ (automatically true in char $0$).

Conclusion: Only the two-dimensional nonabelian algebra ($\lambda \neq 0$), often denoted $\mathfrak{aff}(1)$, is cocomplete; the abelian case $\lambda = 0$ is not.

Dimension 3 ($n=2$): $V = \operatorname{span}\{v_1, v_2\}$, $D \in \mathrm{GL}_2$.

• (a) Diagonal $D = \operatorname{diag}(\lambda_1, \lambda_2)$: must satisfy $\lambda_1, \lambda_2 \neq 0$, $\lambda_1 + \lambda_2 \neq 0$.
• (b) Jordan block $$\begin{pmatrix}\lambda & 1 \ 0 & \lambda\end{pmatrix}$$: $\lambda \neq 0$, $2\lambda \neq 0$.
• (c) Real-complex block $$\begin{pmatrix} a & -b \ b & a \end{pmatrix}$$ (over $\mathbb{R}$): invertibility and non-sum-zero spectral condition.

In each case, the dual requirements of invertibility and absence of eigenvalues summing to zero must be verified. For example, the algebra defined by $\operatorname{diag}(1, -1)$ fails the sum-zero test and is not cocomplete (Le et al., 16 Nov 2025).

6. Duality with Complete Lie Algebras

There is a categorical duality between completeness and cocompleteness in the theory of Lie algebra extensions:

• Completeness: Every cocentral extension (extensions where the kernel maps surjectively onto the quotient of the center) splits trivially. This is equivalent to the vanishing of $H^0(\mathfrak{g},\mathfrak{g})$ and $H^1(\mathfrak{g},\mathfrak{g})$ (trivial center; only inner derivations).
• Cocompleteness: Every central extension splits (quasi-projectivity among Lie algebras). This is equivalent to $H^2(\mathfrak{g}, \mathbb{F}) = 0$.

The classification of almost abelian cocomplete Lie algebras exemplifies the projective-type side of this duality and reinforces the structural correspondences between injective and projective properties in the extension theory of Lie algebras (Le et al., 16 Nov 2025).