Algebras with representable representations

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 $B\to \mathit{Der}(X)$ from $B$ to the Lie algebra $\mathit{Der}(X)$ 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 $\mathbb{K}$, in such a way that these generalised derivations characterise the $\mathbb{K}$-algebra actions. We prove that the answer is no, as soon as the field $\mathbb{K}$ 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 $\mathfrak{gl}(V)$ as a representing object for the representations on a vector space $V$.

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