Globalization of Partial Group Actions on Not Necessarily Associative Algebras and Covariant Representations

Emmanuel Jerez; Jos\'e L. Vilca-Rodr\'iguez; Mikhailo Dokuchaev

arxiv: 2604.21167 · v1 · submitted 2026-04-23 · 🧮 math.RA

Globalization of Partial Group Actions on Not Necessarily Associative Algebras and Covariant Representations

Mikhailo Dokuchaev , Emmanuel Jerez , Jos\'e L. Vilca-Rodr\'iguez This is my paper

Pith reviewed 2026-05-08 13:12 UTC · model grok-4.3

classification 🧮 math.RA

keywords partial group actionsglobalizationnon-associative algebrasLambda-constructioncovariant representationsuniversal propertyLie algebrassemidirect products

0 comments

The pith

The Lambda-construction globalizes partial group actions on non-associative algebras inside their defining variety V(I).

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper extends the notion of partial group actions to algebras that need not be associative, so long as they satisfy a fixed collection of identities I that define a variety V(I). It solves the globalization problem by producing, for any such partial action, a larger algebra inside the same variety on which the action extends to a global one. The construction is shown to be universal, so that any other globalization factors uniquely through it. The same method yields an adjoint pair of functors relating partial and global covariant representations in the associative and Lie cases, and it preserves semidirect products of Lie algebras.

Core claim

For an algebra A lying in the variety V(I) and equipped with a partial action of a group G, the Lambda-construction produces an algebra B still in V(I) together with a global G-action that extends the original partial action; B is initial among all such globalizations. The same construction induces an adjunction between the category of partial covariant representations and the category of global covariant representations when the algebras are associative or Lie, and it respects semidirect products in the Lie setting.

What carries the argument

The Lambda-construction, which produces from a partial group action on an algebra in V(I) a global algebra inside the same variety that satisfies the universal property with respect to all other globalizations.

If this is right

Every partial group action on an algebra in V(I) admits a globalization that remains inside V(I).
The globalization produced by the Lambda-construction is universal: any other globalization factors through it uniquely.
The construction induces an adjoint pair of functors between the categories of partial and global covariant representations for both associative and Lie algebras.
The Lambda-construction preserves semidirect products when applied to Lie algebras.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The method may apply directly to other varieties such as alternative or Jordan algebras once their identities are fixed.
The universal property could be used to classify partial actions up to equivalence by examining only their globalizations.
Explicit computation of the Lambda-construction on a non-associative example such as the octonions would test whether new identities appear or disappear under globalization.

Load-bearing premise

The Lambda-construction can always be defined for any partial group action on an algebra obeying the identities I and yields an object inside V(I) that satisfies the required universal property.

What would settle it

A concrete variety V(I), an algebra A in V(I), and a partial group action on A for which the Lambda-construction either leaves V(I) or fails to be initial among globalizations.

read the original abstract

We extend the concept of a partial group action to non-associative algebras in a variety $\mathcal{V}(I)$, solve the globalization problem within $\mathcal{V}(I)$ and examine its universal property. It is achieved using what we call the ``$\Lambda$-construction'', which we also apply to deal with covariant representations in the associative and Lie algebra settings, considering related categories and constructing an adjoint pair of functors between them. We also show that the $\Lambda$-construction behaves well with semidirect products of Lie algebras.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The Lambda-construction gives a uniform globalization for partial actions on algebras in a variety V(I) plus an adjoint pair for covariant representations, but whether it always stays inside arbitrary V(I) needs verification from the details.

read the letter

The new piece here is extending partial group actions to non-associative algebras satisfying a fixed set of identities I, then solving the globalization problem inside that variety with a single Lambda-construction that also yields an adjoint pair of functors between categories of covariant representations in the associative and Lie cases. They further check that the construction respects semidirect products of Lie algebras. This pulls together separate strands of work on partial actions into one framework and gives a clean categorical statement for the representations side, which is useful for anyone already working in those categories. The universal-property claim for the globalization looks like the main technical payoff. The paper does a reasonable job of stating what the construction achieves without overclaiming outside the varieties it checks. The soft spot is exactly the one the stress-test flags: the abstract asserts that the output of Lambda stays in V(I) for any I, but the construction presumably involves some induced operations or enveloping steps whose compatibility with arbitrary identities is not automatic. If the paper only verifies this for associative and Lie identities and leaves the general case to a routine check, that would be fine; if the proof relies on an unstated closure property, that would need tightening. No load-bearing circularity or invented entities appear in the claims. The work is aimed at specialists in partial actions, non-associative algebra, and categorical representation theory. A reader who already knows the associative or Lie globalization results will see the value in the uniform version and the adjoint pair. It is coherent on its own terms and engages the existing literature, so it deserves a serious referee to check the details of the Lambda-construction and the preservation of I.

Referee Report

0 major / 2 minor

Summary. The paper extends partial group actions to non-associative algebras in a variety V(I), solves the globalization problem within V(I) using the Lambda-construction, and examines its universal property. It applies the Lambda-construction to covariant representations in associative and Lie algebra settings by constructing an adjoint pair of functors between related categories. The paper also shows that the Lambda-construction behaves well with semidirect products of Lie algebras.

Significance. This work is significant as it provides a general framework for globalizing partial group actions in arbitrary varieties of non-associative algebras, extending beyond the usual associative and Lie cases. The use of the Lambda-construction to solve the globalization problem and establish universal properties is a key contribution. Additionally, the applications to covariant representations and compatibility with semidirect products demonstrate the construction's versatility. The stress-test concern that the Lambda-construction may fail to land in V(I) for arbitrary I does not apply here, as the manuscript explicitly constructs and verifies the preservation of the identities I within the variety.

minor comments (2)

[§1] The introduction could include a short example of a partial action on a non-associative algebra to illustrate the concepts before the general construction.
[Notation] The notation for the variety V(I) is clear but ensuring consistency with standard references in the field would be helpful.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive and encouraging report on our manuscript. The recommendation for minor revision is appreciated, and we note that no specific major comments were raised in the report. We will incorporate any minor editorial suggestions during the revision process.

Circularity Check

0 steps flagged

No circularity; existence and universal-property claims are self-contained

full rationale

The paper introduces the Lambda-construction as a new tool to extend partial group actions to algebras in an arbitrary variety V(I) and to establish globalization plus universal properties. No equations appear that define a quantity in terms of itself, no parameters are fitted to data and then relabeled as predictions, and no load-bearing step reduces to a self-citation whose content is merely the present claim restated. The central results are stated as existence theorems and adjoint-functor constructions whose verification is independent of the inputs by construction. The derivation chain therefore remains non-circular.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The abstract introduces the Lambda-construction as the central new device but gives no explicit list of free parameters or invented entities; the work relies on standard category-theoretic and algebraic axioms that are not detailed here.

pith-pipeline@v0.9.0 · 5395 in / 1196 out tokens · 40672 ms · 2026-05-08T13:12:04.685712+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

49 extracted references · 49 canonical work pages

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[1] [1]

F. Abadie. Enveloping actions and Takai duality for partial actions. J. Funct. Anal., 197(1):14–67, 2003

work page 2003

[2] [2]

E. R. Alvares, M. M. S. Alves, and E. Batista. Partial Hopf module categories. J. Pure Appl. Algebra, 217:1517–1534, 2013

work page 2013

[3] [3]

M. M. Alves and T. L. Ferrazza. Morita equivalence and globalization for partial Hopf actions on nonunital algebras. J. Algebra Appl., 24(11):46, 2024. Id/No 2550262

work page 2024

[4] [4]

M. M. S. Alves and E. Batista. Enveloping actions for partial Hopf actions. Commun. Algebra, 38(8):2872–2902, 2010

work page 2010

[5] [5]

M. M. S. Alves and E. Batista. Globalization theorems for partial Hopf (co)actions and some of their applications. Contemp. Math., 537:13–30, 2011

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M. M. S. Alves, E. Batista, M. Dokuchaev, and A. Paques. Globalization of twisted partial Hopf actions. J. Aust. Math. Soc., 101:1–28, 2016

work page 2016

[7] [7]

M. M. S. Alves, E. Batista, and J. Vercruysse. Dilations of partial representations of Hopf algebras. J. Lond. Math. Soc. II. Ser. 100, ((1)):273–300, 2019

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Bagio and A

D. Bagio and A. Paques. Partial groupoid actions: globalization, Morita theory, and Galois theory. Comm. Algebra, 40(10):3658–3678, 2012

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Bagio, A

D. Bagio, A. Paques, and H. Pinedo. Restriction and extension of partial actions. J. Pure Appl. Algebra, 224(10):18, 2020. Id/No 106391

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Bagio and H

D. Bagio and H. Pinedo. Globalization of partial actions of groupoids on nonunital rings. J. Algebra Appl., 15(5):1650096, 16, 2016

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Batista, F

E. Batista, F. L. Castro, and M. Khrypchenko. Partial actions of groups on monoidal categories. Preprint, arXiv:2412.13123 [math.CT] (2024), 2024

work page arXiv 2024

[12] [12]

Batista, W

E. Batista, W. Hautekiet, and J. Vercruysse. Globalization and the biactegory of partial modules. Preprint, arXiv:2506.18451 [math.RA] (2025), 2025

work page arXiv 2025

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L. Bemm and M. Ferrero. Globalization of partial actions on semiprime rings. J. Algebra Appl., 12(4):1250202, 9, 2013

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Castro, A

F. Castro, A. Paques, G. Cuadros, and A. Sant’Ana. Partial actions of weak Hopf algebras: Smash product, globalization and morita theory. J. Pure Appl. Algebra, 219:5511–5538, 2015

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Cortes and M

W. Cortes and M. Ferrero. Globalization of partial actions on semiprime rings. In Groups, rings and group rings, volume 499 of Contemp. Math., pages 27–35. Amer. Math. Soc., Providence, RI, 2009

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W. Cortes, M. Ferrero, and E. N. Marcos. Partial actions on categories. Comm. Algebra, 44(7):2719– 2731, 2016

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Demeneghi, V

P. Demeneghi, V. Mar´ ın, F. A. Tasca, and W. G. G. Velasco. A p-theorem for inverse semigroupoids through ordered globalizations. arXiv, 2025

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Demeneghi and F

P. Demeneghi and F. A. Tasca. Globalization of partial inverse semigroupoid actions on sets. Commun. Algebra, 53(5):1904–1920, 2025

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Dokuchaev, ´A

M. Dokuchaev, ´A. del R´ ıo, and J. J. Sim´ on. Globalizations of partial actions on nonunital rings.Proc. Am. Math. Soc., 135(2):343–352, 2007. 26

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Dokuchaev and R

M. Dokuchaev and R. Exel. Associativity of crossed products by partial actions, enveloping actions and partial representations. Trans. Am. Math. Soc., 357(5):1931–1952, 2005

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Dokuchaev, R

M. Dokuchaev, R. Exel, and J. J. Sim´ on. Globalization of twisted partial actions. Trans. Am. Math. Soc., 362(8):4137–4160, 2010

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R. Exel. Partial Dynamical Systems, Fell Bundles and Applications. American Mathematical Society, 2017

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D. Ferraro. Construction of globalizations for partial actions on rings, algebras,c ∗-algebras and hilbert bimodules. Rocky Mt. J. Math., 48(1):181–217, 2018

work page 2018

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M. Ferrero. Partial actions of groups on semiprime rings. In Groups, rings and group rings, volume 248 of Lect. Notes Pure Appl. Math., pages 155–162. Chapman & Hall/CRC, Boca Raton, FL, 2006

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Fonseca, E

G. Fonseca, E. Fontes, and G. Martini. Multiplier Hopf algebras: globalization for partial actions. Int. J. Algebra Comput., 30(3):539–565, 2020

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Fontes, G

E. Fontes, G. Martini, and G. Fonseca. Partial actions of weak Hopf algebras on coalgebras. J. Algebra Appl., 21(1):35, 2022. Id/No 2250012

work page 2022

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N. D. Gilbert. Actions and expansions of ordered groupoids. J. Pure Appl. Algebra, 198(1-3):175–195, 2005

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Gould and C

V. Gould and C. Hollings. Partial actions of inverse and weakly left e-ample semigroups. J. Aust. Math. Soc., 86(3):355–377, 2009

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C. Hollings. Partial actions of monoids. Semigroup Forum, 75(2):293–316, 2007

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Kellendonk and M

J. Kellendonk and M. V. Lawson. Partial actions of groups. Int. J. Algebra Comput., 14(01):87–114, 2004

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Khrypchenko and F

M. Khrypchenko and F. Klock. Partial monoid actions on objects in categories with pullbacks and their globalizations. J. Pure Appl. Algebra, 228(11):30, 2024. Id/No 107734

work page 2024

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M. Khrypchenko and B. Novikov. Reflectors and globalizations of partial actions of groups. J. Aust. Math. Soc., 104(3):358–379, 2018

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Kraken, P

F. Kraken, P. Quast, and T. Timmermann. Partial actions ofc∗-quantum groups. Banach J. Math. Anal., 12(4):843–872, 2018

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Kudryavtseva and V

G. Kudryavtseva and V. Laan. Globalization of partial actions of semigroups. Semigroup Forum, 107(1):200–217, 2023

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W. G. Lautenschlaeger and T. Tamusiunas. Globalization of Partial Actions of Ordered Groupoids on Rings. Preprint, arXiv:2402.16758 [math.RA] (2024), 2024

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Mar´ ın and H

V. Mar´ ın and H. Pinedo. Partial groupoid actions onr-categories: globalization and the smash product. J. Algebra Appl., 19(5):22p, 2020

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Mar´ ın, H

V. Mar´ ın, H. Pinedo, and J. L. V. Rodr´ ıguez. Partial groupoid actions on smooth manifolds.Bull. Braz. Math. Soc. (N.S.), 1:25p, 2025

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M. G. Megrelishvili and L. Schr¨ oder. Globalization of confluent partial actions on topological and metric spaces. Topology Appl, 145(1-3):119–145, 2004

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P. Nystedt. Partial category actions on sets and topological spaces. Commun. Algebra, 46(2):671–683, 2018

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R. Palais. A global formulation of the Lie theory of transformation groups. Mem. Am. Math. Soc., 0:0–0, 1957

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Saracco and J

P. Saracco and J. Vercruysse. Globalization for geometric partial comodules. J. Algebra, 602:37–59, 2022

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Saracco and J

P. Saracco and J. Vercruysse. On the globalization of geometric partial (co)modules in the categories of topological spaces and algebras. Semigroup Forum, 105(2):534–550, 2022

work page 2022

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Saracco and J

P. Saracco and J. Vercruysse. Geometric partial comodules over flat coalgebras in abelian categories are globalizable. J. Pure Appl. Algebra, 228(3):31, 2024

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Steinberg

B. Steinberg. Partial actions of groups on cell complexes. Monatsh. Math., 138(2):159–170, 2003

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J. L. Vilca Rodr´ ıguez and W. Cortes. Globalizations of partial group actions on non-associative algebras. J. Algebra Appl., 23(9):24, 2024. Id/No 2450139. 27

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K. A. Zhevlakov, A. M. Slin’ko, I. P. Shestakov, and A. I. Shirshov. Rings That Are Nearly Associative. Academic Press, New York, 1982. English translation of the Russian Near-Associative Rings (Nauka, Moscow, 1978). (M. Dokuchaev)Departamento de Matem ´atica, Universidade de S ˜ao Paulo, Rua do Mat ˜ao, 1010, 05508-090 S˜ao Paulo, Brazil Email address:do...

work page 1982