Weak action representability of 2-nilpotent groups

Abstract: In this article, we investigate the representability of actions of the category $\mathsf{Nil}_2(\mathsf{Grp})$ of $2$-nilpotent groups. We first provide an algebraic characterisation of derived actions in $\mathsf{Nil}_2(\mathsf{Grp})$ by determining a universal strict general actor of an object $X$, which turns out to be the group $\operatorname{Aut}_c(X)$ of central automorphisms of $X$. We also characterise the morphisms $B \to \operatorname{Aut}_c(X)$ that define an action of $B$ on $X$ in $\mathsf{Nil}_2(\mathsf{Grp})$. We then show that $\mathsf{Nil}_2(\mathsf{Grp})$ is not action representable, and that the existence of a weak representation is related to the amalgamation property. Using the construction of an amalgam of a suitable family of abelian subgroups of $\operatorname{Aut}_c(X)$, we prove that the category $\mathsf{Nil}_2(\mathsf{Grp})$ is weakly action representable, and that a weak representing object can be chosen to be an abelian group. Finally, we show that $\mathsf{Nil}_2(\mathsf{Grp})$ is not locally algebraically cartesian closed.

• The paper establishes that while the category of 2-nilpotent groups is not strongly action representable, it admits a weak form through abelian groups.
• It provides explicit structure theorems characterizing derived actions via homomorphisms into central automorphism groups.
• It demonstrates that local algebraic cartesian closedness fails in 2-nilpotent groups, contrasting with related semi-abelian categories.

Weak Action Representability in the Category of 2-Nilpotent Groups

Introduction

The paper "Weak action representability of 2-nilpotent groups" (2604.22578) rigorously investigates the (weak) action representability properties of the category $\_2(\mathsf{Gp})$ of 2-nilpotent groups within the framework of semi-abelian categories and categorical algebra. The work draws on recent advances in the characterization of internal actions, actor theory, and algebraic amalgamation, motivated in part by existing results on categories such as Lie algebras and associative algebras and their subvarieties. The paper provides authoritative algebraic and categorical results, includes explicit structure theorems, and addresses several open problems regarding representability properties and their interaction with conditions such as local algebraic cartesian closedness.

Construction and Characterization of Actions in $\_2(\mathsf{Gp})$

The authors provide an explicit description of actions in $\_2(\mathsf{Gp})$ by analyzing split extensions and derived actions. The derived action of a 2-nilpotent group $B$ on $X$ is characterized by a group homomorphism $B \rightarrow \Aut_c(X)$, where $\Aut_c(X)$ denotes the central automorphisms of $X$, i.e., automorphisms $f$ such that $f(x)x^{-1}\in \bZ(X)$ for all $\_2(\mathsf{Gp})$0. The construction delineates which morphisms $\_2(\mathsf{Gp})$1 correspond to legitimate actions: those for which

$\_2(\mathsf{Gp})$2

for all $\_2(\mathsf{Gp})$3 and $\_2(\mathsf{Gp})$4. This ensures that the $\_2(\mathsf{Gp})$5-orbit of the defect $\_2(\mathsf{Gp})$6 is trivial, and leads to a strong constraint: the image of $\_2(\mathsf{Gp})$7 is necessarily an abelian subgroup of $\_2(\mathsf{Gp})$8.

An important structural result is that the universal strict general actor (USGA) of a 2-nilpotent group $\_2(\mathsf{Gp})$9 in the sense of Casas et al. is precisely $\_2(\mathsf{Gp})$0.

The authors also provide explicit counterexamples demonstrating that not every group morphism $\_2(\mathsf{Gp})$1 yields a split extension in $\_2(\mathsf{Gp})$2, due to violations of the above constraint or the resulting semidirect product exceeding the required nilpotency class.

On (Weak) Action Representability and the Amalgamation Property

The notion of action representability is categorically stringent. The authors recall that $\_2(\mathsf{Gp})$3 is an Orzech category of interest and hence action accessible, but not action representable: for many $\_2(\mathsf{Gp})$4, $\_2(\mathsf{Gp})$5 is not 2-nilpotent and so cannot serve as a representing object [(2604.22578), Prop. 3.1]. The functor sending $\_2(\mathsf{Gp})$6 to the set of actions on $\_2(\mathsf{Gp})$7 admits only an injective (not bijective) transformation to $\_2(\mathsf{Gp})$8.

However, the main contribution is the establishment of weak action representability for $\_2(\mathsf{Gp})$9. Building on recent work in the context of $B$0-nilpotent Lie algebras, the authors show that for each $B$1, there exists an abelian group $B$2 and a natural monomorphism

$B$3

such that $B$4 is a weak representing object for the functor of actions. The key tool is the existence of the amalgamation property (AP) for abelian groups: the family of abelian subgroups $B$5 (the images of the corresponding action-defining morphisms) can be amalgamated into a single abelian group $B$6, and each action is encoded via a canonical map to this $B$7. This leverages the algebraic structure of abelian groups and the behavior of actions in $B$8.

A strong claim of the paper is that, in contrast to higher nilpotency classes ($B$9), where weak action representability fails due to the breakdown of the amalgamation property for the relevant categories, for the case $X$0 this amalgamation ensures weak representability. The result closes an open question from prior work [(2604.22578), Thm. 4.4].

Failure of Local Algebraic Cartesian Closedness

The final part of the article addresses local algebraic cartesian closedness (LACC) for $X$1. LACC is characterized by the preservation of certain kernel structures under coproducts. The authors exhibit an explicit failure: the canonical comparison morphism

$X$2

is not (in general) a monomorphism for appropriate choices of $X$3 in $X$4, as demonstrated via Heisenberg group calculations. Therefore, $X$5 is not locally algebraically cartesian closed—a property otherwise characteristic of categories such as groups and Lie algebras.

This result implies that LACC is strictly stronger than weak action representability and action accessibility in this context. The negative result for LACC also contributes to the ongoing question of the precise categorical relationship between various representability and algebraic closedness properties in non-abelian and restricted algebraic categories.

Practical and Theoretical Implications

The identification of the structure of actors and weak actors for 2-nilpotent groups has direct implications for the representation theory of nilpotent groups, cohomology theories, and the classification of extensions in such categories. Practically, explicit weak representing objects facilitate calculations and constructions involving split extensions and derivations. The explicit failure of LACC restricts the range of categorical techniques transferable from group theory and Lie theory to 2-nilpotent groups and highlights nuanced differences even between apparently similar algebraic varieties.

On the theoretical side, the characterization of action representability and its weak form enhances understanding of internal categorical structures, commutators, and the relationship between amalgamation and representation properties. The analogy with 2-nilpotent Lie algebras and the contrast with higher nilpotency classes clarifies the boundaries of these categorical phenomena and suggests that further investigation of (non-)representability in related subvarieties could yield new categorical invariants or obstructions.

Future developments may include a more detailed analysis of weak representing objects in other varieties of nilpotent groups, connections with algebraic exponentiation, and the creation of categorical frameworks accommodating both the combinatorics of amalgamation and the algebraic constraints inherent in nilpotency.

Conclusion

This paper provides a comprehensive categorical analysis of action representability for 2-nilpotent groups, revealing that while the category is not action representable in the strong sense, it is weakly action representable, with abelian groups serving as weak representing objects. The precise algebraic conditions connecting actions with morphisms into the group of central automorphisms, and the explicit utilization of the amalgamation property, lead to a fine-grained understanding of the representability landscape for nilpotent group categories. The work further establishes that $X$6 is not locally algebraically cartesian closed, situating its place within the spectrum of semi-abelian categories and informing the ongoing study of action representability and categorical properties in non-abelian algebraic settings.