Algebras with representable representations

Abstract: (Joint work with Xabier García-Martínez, Matsvei Tsishyn and Corentin Vienne)

Just like group actions are represented by group automorphisms, Lie algebra actions are represented by derivations: up to isomorphism, a split extension of a Lie algebra B by a Lie algebra X corresponds to a Lie algebra morphism B$\to\mathbf{Der}$(X) from B to the Lie algebra $\mathbf{Der}$(X) of derivations on X. The aim of this talk is to elaborate on the question, whether the concept of a derivation can be extended to other types of non-associative algebras over a field $\mathbf{K}$, in such a way that these generalised derivations characterise the $\mathbf{K}$-algebra actions. We prove that the answer is no, as soon as the field $\mathbf{K}$ is infinite. In fact, we prove a stronger result: already the representability of all abelian actions – which are usually called representations or Beck modules – suffices for this to be true. Thus we characterise the variety of Lie algebras over an infinite field of characteristic different from 2 as the only variety of non-associative algebras which is a non-abelian category with representable representations. This emphasises the unique role played by the Lie algebra of linear endomorphisms $\mathbf{gl}$(V) as a representing object for the representations on a vector space V.

category theory

Audience: researchers in the topic

ItaCa Fest 2021

Series comments: ItaCa Fest is an online webinar aimed to gather the community of ItaCa.