Cocentral Extensions in Algebra

• Cocentral extensions are algebraic constructs dual to central extensions, defined by an abelian quotient and non-central kernel behavior.
• They play a crucial role in both Lie and Hopf algebra theory by enabling cohomological classifications and identifying splitting properties via vanishing second cohomology.
• Applications include classifying almost abelian Lie algebras and constructing non-pointed Hopf algebras using graded twistings and explicit group cohomological data.

A cocentral extension is a central concept in the theory of algebraic structures, particularly Lie algebras and Hopf algebras, arising as the dual counterpart to central extensions. In a cocentral extension, the extension sequence is characterized by the abelianity of the quotient (as opposed to the centrality of the kernel in central extensions), with the coalgebraic or Lie-algebraic image lying in a “center” in the dual sense. Cocentral extensions appear fundamentally in the structure theory, cohomological classification, and categorical dualities of both Lie algebras and Hopf algebras, and underpin the projective-like properties of cocomplete algebras and the multiplicativity behavior of coideal subalgebras.

1. Cocentral Extensions in Lie Algebras

A short exact sequence of Lie algebras

$$0 \to A \xrightarrow{i} B \xrightarrow{\pi} V \to 0$$

is called a cocentral extension if $V$ is abelian (i.e., $[v,w]=0$ for all $v,w\in V$) and the image $i(A)$ may be interpreted as being “maximal non-central” in $B$.

This definition stands in duality to that of a central extension, where the kernel is abelian and lies in the center of $B$. Cocentral extensions are significant for the classification of Lie algebras which split trivially over any abelian quotient, a property termed completeness. Conversely, the projective-like dual is the notion of cocompleteness, which means that all central extensions by abelian $V$ split trivially; cocompleteness is characterized cohomologically by the vanishing of the second Lie algebra cohomology group: $$\mathfrak{g} \text{ cocomplete} \iff H^2(\mathfrak{g},\Bbb K) = 0$$ (Le et al., 16 Nov 2025).

In this setting, cocentral extensions arise as the categorical duals to central extensions, permitting a dual framework in the structure theory of Lie algebras, where injectivity is replaced by projectivity, and completeness by cocompleteness.

2. Cocentral Extensions and Cocompleteness: Cohomological Aspects

Cocentral extensions play a pivotal role in the cohomological characterization of cocomplete Lie algebras. Given a Lie algebra $\mathfrak{g}$, the splitting property for all central extensions is governed by $H^2(\mathfrak{g},\Bbb K)$. Vanishing cohomology guarantees the existence of a splitting homomorphism in every central extension, thereby yielding a cocomplete structure.

In almost abelian cases, for $\mathfrak{g} = \Bbb K^n \oplus_D \Bbb K e_0$, Hochschild–Serre computations yield: $$H^2(\mathfrak{g},\Bbb K) \cong \Lambda^2_D((\Bbb K^n)^*) \oplus ((\Bbb K^n)^* / \mathrm{Im}(D^T))$$ where cocompleteness is equivalent to $D$ invertible and no pair of eigenvalues summing to zero (Le et al., 16 Nov 2025).

Semisimple Lie algebras are both complete and cocomplete ($H^1=H^2=0$ by Whitehead's Lemmas), while nilpotent Lie algebras fail both properties.

3. Cocentral Extensions in Hopf Algebras

In Hopf algebra theory, a cocentral extension

$B \xrightarrow{i} A \xrightarrow{\pi} Q$

is defined such that $\pi$ is surjective and cocentral, i.e.,

$$\pi(a_{(1)}) \otimes a_{(2)} = \pi(a_{(2)}) \otimes a_{(1)} \quad \forall\;a \in A$$

(Bichon et al., 2020). When $B$ is commutative, the structure is termed an abelian cocentral extension.

Cocentral extensions facilitate graded twistings of function algebras and the formation of Kac algebras. For $A = k^G\;^{\tau}\!\#_{\sigma}kF$, $G,F$ finite groups, the extension data comprises:

• An $F$-action on $G$
• Twisted 2-cocycle $\sigma:F\times F \to k^G$
• Dual cocycle $\tau:F \to k^G \otimes k^G$

These equip the Hopf algebra with multiplicative and coalgebraic structures defining the cocentral abelian extension (Burciu, 2012).

4. Classification and Isomorphism Problems for Cocentral Extensions

Classification of cocentral (particularly abelian) extensions leverages combinatorial and cohomological data, notably “$m$-data” for cyclic group gradings. Let $G$ be a finite group and $\Gamma = C_m$ a cyclic group; the graded twisting $O(G)^{i,\alpha}$ is determined by:

• An injective map $i:\Gamma \to Z(G)$ (central embedding)
• Automorphism $\alpha:\Gamma \to \Aut(G)$ fixing $i(\Gamma)$

The isomorphism class is governed by equivalence of the associated $m$-data $(H,\theta,a,T)$, encoding the quotient group, automorphism of $H$, scalar function, and cocycle (Bichon et al., 2020). Explicit enumeration is possible for several families such as dihedral, symmetric, and alternating groups.

Hopf algebraic analogues of the Wall conjecture for groups have recently been verified in the cocentral Kac algebra case: the number of maximal coideal subalgebras does not exceed the dimension of $A$ (Burciu, 2012).

5. Structure and Examples: Cleft and Cocentral Extensions in Hopf Theory

Cocentral abelian cleft extensions feature prominently in the structure theory of semisimple Hopf algebras. For instance, the unique nontrivial semisimple Hopf algebra of dimension 8 (Kac-Paljutkin algebra) is constructed via cocentral abelian extension data ($G=C_2 \times C_2$, $F=C_2$ with nontrivial cocycle).

Extensions of the form

$1 \to k^N \to K_n \to kQ \to 1$

with $N=C_n \times C_n$ and $Q=C_2$ yield the semisimple involutive Hopf algebra $K_n$ as a cocentral abelian cleft extension (Garcia et al., 2023).

Nichols algebras, fusion rules, and bosonization constructions frequently exploit such extensions, leading to the construction of new non-pointed Hopf algebras (e.g., dimension 216 algebras via bosonization of Fomin-Kirillov Nichols algebra over $K_3$) (Garcia et al., 2023).

6. Duality Framework: Completeness and Cocompleteness

Cocentral extensions and cocomplete algebras instantiate a duality in the extension theory of Lie algebras and analogous phenomena in Hopf algebras. The key dimensions are:

Property	Central Extensions (Injective)	Cocentral Extensions (Projective)
Object	Complete Lie algebra ($Z=0$,$\Der=\ad$)	Cocomplete Lie algebra ($H^2=0$)
Splitting	Splits any cocentral extension trivially	Splits any central extension trivially
Cohomology	$H^0=0$, $H^1=0$	$H^2=0$
Exemplars	Semisimple Lie algebras	Semisimple Lie algebras

Nilpotent Lie algebras fail both; no Lie algebra is injective or projective for arbitrary extensions (Le et al., 16 Nov 2025).

This duality framework extends the classical module theory into structural Lie and Hopf algebra settings, facilitating deeper comprehension of splitting phenomena and the underlying algebraic infrastructure.

7. Applications and Classification Results

Concrete classification for cocentral extensions is available for various group and algebra families:

• Graded twists of $O(G)$ by cyclic $\Gamma$ (determined by central embeddings and automorphisms)
• Enumeration of noncommutative abelian cocentral extensions for dihedral, alternating, and symmetric groups with explicit counts for each family (Bichon et al., 2020)
• For cocentral abelian cleft extensions with coradical $K_n$, classification of Yetter–Drinfeld modules, fusion rules, and Nichols algebras (Garcia et al., 2023)
• Explicit classification of almost abelian cocomplete Lie algebras by proportional similarity of derivations, with canonical representatives provided [(Le et al., 16 Nov 2025), Tables 4.11–4.12]

A plausible implication is that cocentral extension theory provides the categorical infrastructure for projective and splitting behaviors in noncommutative algebra, with wide-ranging consequences in both finite group theory and quantum algebra.