Algebras with representable representations
(2021) Edinburgh Mathematical Society. Proceedings β Vol. 64, p. 555-573 (2021)
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- 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 BβDer(X) from B to the Lie algebra Der(X) of derivations on X. In this article, we study the question whether the concept of a derivation can be extended to other types of non-associative algebras over a field π, in such a way that these generalised derivations characterise the π-algebra actions. We prove that the answer is no, as soon as the field π 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 π€π©(V) as a representing object for the representations on a vector space V.
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GarcΓa MartΓnez, X., Tsishyn, M., Van der Linden, T., & Vienne, C. (2021). Algebras with representable representations. Edinburgh Mathematical Society. Proceedings, 64, 555-573. https://doi.org/10.1017/S0013091521000304 (Original work published 2021)