Action accessible and weakly action representable varieties of algebras
(2026) Bollettino dell’Unione Matematica Italiana — Vol. 19, n° 2, p. 367-374 (2026)
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- Abstract
- (en) 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 3) and the varieties of n-solvable Lie algebras (n 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 3) and that of 2-solvable groups are not weakly action representable.
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García-Martínez, X., & Mancini, M. (2026). Action accessible and weakly action representable varieties of algebras. Bollettino dell’Unione Matematica Italiana, 19(2), 367-374. https://doi.org/10.1007/s40574-025-00496-1 (Original work published 2026)