Weak representability of actions of non-associative algebras
(2023) 4th ItaCa Workshop — Location: Turin, Italy (18.December.2023)
Files
ItaCa Workshop 2024 - Slides Manuel Mancini.pdf
Open Access - Adobe PDF
- 330.58 KB
Details
- Authors
- Abstract
- (en) It is well known that, in the semi-abelian category Lie of Lie algebras over a field F, algebra actions are represented by derivations. This means that the category Lie is action representable and that the representing object, called the actor, is the Lie algebra of derivations. The notion of an action representable category has proven to be quite restrictive. For example, if a non-abelian variety V of non-associative algebras over an infinite field F, with char(F) ≠ 2, is action representable, then V must be Lie. More recently, G. Janelidze introduced the notion of a weakly action representable category, which includes a wider class of categories, such as the variety Assoc of associative algebras and the variety Leib of Leibniz algebras. In this talk we show that the converse of the implication Weakly Action Representable Category ⇒ Action Accessible Category is false also in the context of varieties of non-associative algebras. Then, for an algebraically coherent and operadic variety V and an object X of V, we show that it is always possible to construct a partial algebra E(X), called the external weak actor of X, together with a monomorphism of functors τ : Act(-, X) → Hom_PAlg(-, E(X)), where PAlg is the category of partial algebras over F. The pair (E(X), τ) is called the external weak representation of the functor Act(-, X). Moreover, for any other object B in V, we provide a complete description of the morphisms B → E(X) belonging to the image of τ_B, that is, the homomorphisms of partial algebras that identify the actions of B on X in V. This is joint work with Xabier García Martínez (Universidade de Vigo, Spain), Tim Van der Linden and Corentin Vienne (Université catholique de Louvain, Belgium).
- Affiliations
Citations
Mancini, M. (2023, December 18). Weak representability of actions of non-associative algebras. 4th ItaCa Workshop, Turin, Italy. https://hdl.handle.net/2078.5/279074