103. 20/01/2025 Manuel Mancini (University of Palermo, Italy)

Title: On the representability of actions of non-associative algebras
Abstract: It is well known that in the semi-abelian category Grp of groups, split extensions, or equivalently internal actions, are represented by automorphisms. This means that the category Grp is action representable and the actor of a group X is the group Aut(X). The notion of action representable category has proven to be quite restrictive: for instance, if a non-abelian variety of non-associative algebras, over an infinite field of characteristic different from two, is action representable, then it is the category of Lie algebras. More recently G. Janelidze introduced the notion of weakly action representable category, which includes a wider class of categories. In this talk we show that for an algebraically coherent variety of algebras and an object X of it, it is always possible to construct a partial algebra E(X), called external weak actor of X, which allows us to describe internal actions on X. Moreover, we show that the existence of a weak representation is connected to the amalgamation property, and we give an application of the construction of the external weak actor in the context of varieties of unitary algebras. This is joint work with J. Brox (Universidad de Valladolid), Xabier García Martínez (Universidade de Vigo), Tim Van der Linden and Corentin Vienne (Université catholique de Louvain).
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