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.