Algebras with representable representations

Just like group actions are represented by automorphisms, Lie algebra derivations: up to isomorphism, a split extension of $B$ $X$ corresponds morphism $B\to \mathit{Der}(X)$ from the $\mathit{Der}(X)$ derivations on $X$. In this article, we study question whether concept derivation can be extended other types non-associative algebras over field $\mathbb{K}$, in such way that these generalised characterise $\mathbb{K}$-algebra actions. We prove answer is no, as soon $\mathbb{K}$ infinite. fact, stronger result: already representability all abelian -- which usually called representations or Beck modules suffices for true. Thus variety an infinite characteristic different $2$ only non-abelian category with representable representations. This emphasises unique role played linear endomorphisms $\mathfrak{gl}(V)$ representing object vector space $V$.