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.