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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 $V$
.
