Actions in varieties of non-associative algebras: a survey

Abstract

This article surveys the main results on the representability of internal actions in varieties of non-associative algebras over a field. We begin by recalling the general framework of internal actions and split extensions in semi-abelian categories, before specialising to Orzech categories of interest and varieties of non-associative algebras. We discuss action representability, weak action representability, and action accessibility, with emphasis on the relationship between the latter two notions. We then describe the constructions of the universal strict general actor in Orzech categories of interest and the external weak actor in action accessible, operadic varieties of non-associative algebras over a field, as well as their role in the study of weak action representability. We also give a tensor-domain formulation of partial algebras and adapt the construction of the external weak actor to this setting. These notions are illustrated by examples arising from associative, Lie, Leibniz, Jacobi-Jordan, anti-commutative anti-associative, alternative, and $2$-nilpotent algebras. We conclude with a summary of the known results and some open questions for future research.

Varieties of associative algebras with quadratic codimension growth
Wesley Quaresma Cota, Luiz Henrique de Souza Matos, Pedro Quintino
arXiv14 Sep 2026PreprintCC BY 4.0Full textAlgebra & number theoryarXiv
Varieties of associative algebras with slow central codimension growth
Antonio Ioppolo, Wesley Quaresma Cota, Ana Cristina Vieira
arXiv1 Nov 2026PreprintCC BY 4.0Full textAlgebra & number theoryarXiv
A Non-linearizable Action of S₃ on Free Metabelian Lie and Associative Algebras
Altyngul Naurazbekova, Ualbai Umirbaev
arXiv19 Oct 2026PreprintCC BY 4.0Full textAlgebra & number theoryarXiv
The Parafree Conjecture for Lie rings and associative algebras
Vasily Ionin, Roman Mikhailov
arXiv19 Oct 2026PreprintCC0Full textAlgebra & number theoryarXiv
Derived Associative Algebras and Cyclic Cohomology of Q-manifolds
Andrew James Bruce
arXiv4 Sep 2026PreprintarXiv licenceFull textAlgebra & number theory Geometry & topologyarXiv

Cite

BibTeX RIS

BibTeX

@misc{mancini2026actions,
  title = {{Actions in varieties of non-associative algebras: a survey}},
  author = {Manuel Mancini},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.18302},
  eprint = {2610.18302},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.18302},
  note = {CC BY-SA 4.0},
}

RIS

TY  - UNPB
TI  - Actions in varieties of non-associative algebras: a survey
AU  - Manuel Mancini
PY  - 2026
DA  - 2026/10/13
PB  - arXiv
DO  - 10.48550/arxiv.2610.18302
UR  - https://arxiv.org/abs/2610.18302
AB  - This article surveys the main results on the representability of internal actions in varieties of non-associative algebras over a field. We begin by recalling the general framework of internal actions and split extensions in semi-abelian categories, before specialising to Orzech categories of interest and varieties of non-associative algebras. We discuss action representability, weak action representability, and action accessibility, with emphasis on the relationship between the latter two notions. We then describe the constructions of the universal strict general actor in Orzech categories of interest and the external weak actor in action accessible, operadic varieties of non-associative algebras over a field, as well as their role in the study of weak action representability. We also give a tensor-domain formulation of partial algebras and adapt the construction of the external weak actor to this setting. These notions are illustrated by examples arising from associative, Lie, Leibniz, Jacobi-Jordan, anti-commutative anti-associative, alternative, and $2$-nilpotent algebras. We conclude with a summary of the known results and some open questions for future research.
ER  -

Abstract

Related

Cite