Action accessible and weakly action representable varieties of algebras

Abstract

The main goal of this article is to investigate the relationship between action accessibility and weak action representability in the context of varieties of non-associative algebras over a field. Specifically, using an argument of J. R. A. Gray in the setting of groups, we prove that the varieties of $k$-nilpotent Lie algebras ($k \geq 3$) and the varieties of $n$-solvable Lie algebras ($n \geq 2$) do not form weakly action representable categories. These are the first known examples of action accessible varieties of non-associative algebras that fail to be weakly action representable, establishing that a subvariety of a (weakly) action representable variety of non-associative algebras needs not be weakly action representable. Eventually, we refine J. R. A. Gray's result by proving that the varieties of $k$-nilpotent groups ($k \geq 3$) and that of $2$-solvable groups are not weakly action representable.