Non-abelian Extensions of Lie algebras with derivations

Abstract: In this paper, we investigate non-abelian extensions of Lie algebras with derivations using several different approaches. We show that the theory of non-abelian extensions of a Lie algebra with a derivation can be characterized by means of the second non-abelian cohomology, the Deligne groupoid, the homotopy category of strict Lie $2$-algebras with strict derivations, and the notion of a $(\g, D)$-kernel, respectively. Moreover, within this unified framework, we address the following existence problem: given a non-abelian extension of Lie algebras [\begin{CD} 0@>>>\h@>i>>\hat{\g}@>p>>\g @>>>0, \end{CD}] let $(K,D)\in\Der(\h)\times\Der(\g)$ be a pair of derivations of $\h$ and $\g$ respectively. When does there exist a derivation $\hat{D}$ of $\hat{\g}$ such that $\hat{D}|_\h=K$ and $D\circ p=p\circ\hat{D}.$ We provide an obstruction class for the existence of such a lift.

• The paper presents a cohomological construction identifying non-abelian extensions by classifying LieDer pairs via second non-abelian cohomology.
• It employs differential graded Lie algebras and Deligne groupoids to connect extension theory with deformation and homotopy frameworks.
• The work provides explicit cohomological criteria and obstruction classes to determine the extensibility of derivations in non-abelian settings.

Non-Abelian Extensions of Lie Algebras with Derivations: A Cohomological and Categorical Framework

Introduction and Main Results

The paper "Non-abelian Extensions of Lie algebras with derivations" (2604.26276) presents a comprehensive treatment of non-abelian extensions of Lie algebras equipped with derivations (“LieDer” pairs). This work constructs a unified algebraic, cohomological, and higher-categorical framework for understanding, classifying, and analyzing the equivalence, existence, and obstruction theory for non-abelian extensions of LieDer pairs.

Four conceptual frameworks are integrated:

1. Second non-abelian cohomology for LieDer pairs, providing a cohomological classification.
2. Deligne groupoids and differential graded Lie algebras (dgLa), linking extension theory to deformation-theoretic structures.
3. The homotopy category of strict Lie $2$-algebras with strict derivations, relating cohomological classification to categorical representations.
4. The notion of $(\mathfrak{g}, D)$-kernels, connecting kernels, representations, and obstruction theory.

Crucially, the paper addresses the extensibility of derivations through non-abelian extensions, characterizing obstructions via explicit cohomological invariants.

Cohomological Classification of Non-Abelian Extensions

Given LieDer pairs $(\mathfrak{g}, D)$ and $(\mathfrak{k}, K)$, a non-abelian extension is a short exact sequence of LieDer pairs

$$0 \to (\mathfrak{k}, K) \to (\hat{\mathfrak{g}}, \hat{D}) \to (\mathfrak{g}, D) \to 0$$

where all structure maps are compatible with the respective derivations. The central problem is to classify, up to isomorphism, such extensions.

The classification is achieved using non-abelian $2$-cocycles: triples $(\varrho, \omega, \chi)$ encoding an action, a bracket-twisting, and a derivation-twisting, which satisfy compatibility equations generalizing Lie algebra cohomology to the non-abelian setting and incorporating derivations. The set of equivalence classes of these non-abelian $2$-cocycles is identified with the second non-abelian cohomology group $H_{nab}^2(\mathfrak{g}, D ; \mathfrak{k})$.

The main classification theorem proves a bijection between isomorphism classes of non-abelian extensions and $H_{nab}^2(\mathfrak{g}, D ; \mathfrak{k})$.

Categorical and Homotopical Perspectives

Deligne Groupoid Interpretation

The authors construct a differential graded Lie algebra $(\mathfrak{g}, D)$0 whose Maurer-Cartan (MC) elements correspond to non-abelian $(\mathfrak{g}, D)$1-cocycles, and whose Deligne groupoid encodes gauge equivalence among MC elements. They show:

• The set of isomorphism classes of non-abelian extensions is in bijection with $(\mathfrak{g}, D)$2 of the Deligne groupoid, i.e., with gauge-equivalence classes of MC elements.
• This perspective connects non-abelian extension classification to deformation theory and higher category structures.

Lie 2-Algebra and Homotopy Category Approach

A correspondence is established between non-abelian extensions of $(\mathfrak{g}, D)$3 by $(\mathfrak{g}, D)$4 and homotopy classes in the category of strict Lie $(\mathfrak{g}, D)$5-algebras with strict derivations. The functor

$(\mathfrak{g}, D)$6

is shown to be representable in the homotopy category of LieDer pairs, with the representing object constructed explicitly from derivation data.

$(\mathfrak{g}, D)$7-Kernels, Cohomology, and Obstructions

The paper introduces the notion of a $(\mathfrak{g}, D)$8-kernel for $(\mathfrak{g}, D)$9, a homomorphism from $(\mathfrak{g}, D)$0 into the outer derivations of $(\mathfrak{g}, D)$1, modulo compatibility with $(\mathfrak{g}, D)$2 and $(\mathfrak{g}, D)$3. This kernel controls the possibility of extending $(\mathfrak{g}, D)$4 by $(\mathfrak{g}, D)$5 and leads to a refined parameterization:

• For a fixed integrable kernel, isomorphism classes of extensions correspond bijectively to the second cohomology group $(\mathfrak{g}, D)$6, measured via a representation induced by the kernel.
• The integrability of a kernel—i.e., the existence of a corresponding extension—is governed by the vanishing of an explicitly constructed cohomology class in $(\mathfrak{g}, D)$7.

This fully resolves the existence and parameterization problems for non-abelian extensions of LieDer pairs.

Extensibility of Derivations and Associated Obstructions

The extensibility problem asks: Given a non-abelian extension $(\mathfrak{g}, D)$8 and derivations $(\mathfrak{g}, D)$9, $(\mathfrak{k}, K)$0, does there exist $(\mathfrak{k}, K)$1 interpolating $(\mathfrak{k}, K)$2 and $(\mathfrak{k}, K)$3 in the extension?

The results:

• $(\mathfrak{k}, K)$4 is extensible iff there exists a solution to a system of compatibility equations, i.e., $(\mathfrak{k}, K)$5 and $(\mathfrak{k}, K)$6 are compatible and an associated cohomology class $(\mathfrak{k}, K)$7 vanishes in $(\mathfrak{k}, K)$8.
• The map $(\mathfrak{k}, K)$9 provides a concrete obstruction class: extensibility is obstructed precisely by the non-vanishing of $$0 \to (\mathfrak{k}, K) \to (\hat{\mathfrak{g}}, \hat{D}) \to (\mathfrak{g}, D) \to 0$$0.

In the central extension case, this reduces to existing obstruction-theoretic statements. This result is highly explicit, yielding practical cohomological criteria for derivation extensibility in non-abelian contexts.

Implications and Future Directions

This work synthesizes non-abelian cohomology, higher Lie theory, and obstruction theory for Lie algebras with derivations. The integration of cohomological, categorical, and deformation-theoretic approaches is technically significant:

• It provides a blueprint for extension theory in the presence of additional algebraic structure (e.g., derivations, modules over other operads).
• The categorical models (strict Lie $$0 \to (\mathfrak{k}, K) \to (\hat{\mathfrak{g}}, \hat{D}) \to (\mathfrak{g}, D) \to 0$$1-algebras, Deligne groupoids) invite further investigation into higher and homotopical generalizations (e.g., $$0 \to (\mathfrak{k}, K) \to (\hat{\mathfrak{g}}, \hat{D}) \to (\mathfrak{g}, D) \to 0$$2-algebras with derivations, stacks of LieDer pairs).
• The obstruction theory developed here extends the scope of classical Lie algebra extension theory. Potential future developments include applications to integration problems for Lie groupoids with additional structure, deformation-quantization, or canonical quantizations of non-abelian gerbes.

Conclusion

The paper establishes a robust, multi-faceted framework for non-abelian extensions of Lie algebras with derivations, unifying cohomological classification, higher categorical descriptions, and explicit obstruction theory. This treatment clarifies the extensibility of derivations and advances the structure theory of Lie algebras with additional derivational structure, providing foundational tools for further work in homological and higher algebra in Lie theoretic settings.