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 $B\to \mathrm{Der}(X)$ from $B$ to the Lie algebra $\mathrm{Der}(X)$ 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 $K$, in such a way that these generalised derivations characterise the $K$-algebra actions. We prove that the answer is~no, as soon as the field $K$ is infinite. In fact, we prove a stronger result: already the representability of all \emph{abelian} actions---which are usually called \emph{representations} or \emph{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 $\mathrm{gl}(V)$ as a representing object for the representations on a vector space~$V$.