Action accessible and weakly action representable varieties of algebras

Mancini, Manuel
(2025) LAC: Logic, Algebra, and Categories Seminar — Location: Milan, Italy (2025)

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The aim of this talk is to investigate the relationship between action accessibility and weak action representability in the context of varieties of non-associative algebras over a field. Using an argument of J. R. A. Gray in the setting of groups, we prove that the varieties of k-nilpotent Lie algebras (k>2) and the varieties of n-solvable Lie algebras (n>1) are not 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 need not be weakly action representable. If time permits, 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. This is joint work with Xabier García-Martínez (Universidade de Vigo, Spain).
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Mancini, M. (2025, March 27). Action accessible and weakly action representable varieties of algebras. LAC: Logic, Algebra, and Categories Seminar, Milan, Italy. https://hdl.handle.net/2078.5/279068