Weak Representability of Actions of Non-Associative Algebras
(2023) PSSL 107: Peripatetic Seminar on Sheaves and Logic — Location: Athens, Greece (1.April.2023)
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- Abstract
- (en) It is well known that, in the semi-abelian category LieAlg₍F₎ of Lie algebras over a field F with char(F) ≠ 2, algebra actions are represented by derivations. This means that the category LieAlg₍F₎ 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 is action representable, then V must be the category LieAlg₍F₎. More recently, G. Janelidze introduced the notion of a weakly action representable category, which includes a wider class of categories, such as the variety AAlg₍F₎ of associative algebras and the variety LeibAlg₍F₎ of Leibniz algebras. In this talk we answer one of the open questions formulated by G. Janelidze. 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 [X] and a monomorphism of functors τ : Act(-, X) → Hom₍PAlg₍F₎₎(-, [X]), where PAlg₍F₎ is the category of partial algebras over F. Moreover, for any other object B in V, we provide a complete description of the morphisms B → [X] belonging to the image of τ_B, i.e., 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), Tim Van der Linden and Corentin Vienne (Université catholique de Louvain).
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Mancini, M. (2023, April 1). Weak Representability of Actions of Non-Associative Algebras. PSSL 107: Peripatetic Seminar on Sheaves and Logic, Athens, Greece. https://hdl.handle.net/2078.5/279072