Federica PIAZZA (Università degli Studi di Messina) and Manuel MANCINI (Università degli Studi di Palermo) will give talks, respectively, on
On split extensions of hoops
Abstract:
Internal actions have been introduced by F. Borceux, G. Janelidze, and G. M. Kelly in order to generalize the equivalence between actions and split extensions from the category of groups to more general settings, e.g. semi-abelian categories. However, in several cases, internal actions may be expressed in terms of so-called external actions, i.e. via set-theoretical maps satisfying a specified set of identities.
In this talk, we will study the relationship between external actions and split extensions in the category Hoops of hoops. In particular, we will focus on those split extensions which strongly splits, i. e. such that the corresponding split epimorphism has a strong section, as introduced by Rump [1] in order to obtain a description of the semidirect products in the category of L-algebras, and compare this with the semidirect product constructions, as described by Clementino, Montoli and Sousa in the more general context of semi-abelian categories [2].
This is joint work with M. Mancini, G. Metere and M. E. Tabacchi.
[1] W. Rump, The category of L-algebras, TAC 39, No. 21, 2023.
[2] M. M. Clementino, A. Montoli, L. Sousa, Semidirect products of (topological) semi-abelian algebras, JPAA 219, No. 1, 2015.
and
Action accessible and weakly action representable varieties of algebras
Abstract:
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).