2025/03/21 by García-Martínez, Xabier, Mancini, Manuel · 2 citations
#08A35 #08C05 #16W25 #17A36 #18E13 #Category Theory (math.CT) #FOS: Mathematics #Rings and Algebras (math.RA)
paper · doi:10.48550/arxiv.2503.17326
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.