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 generalized derivations characterize 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 characterize 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 emphasizes 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 $V$ .