  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 $V$.
