Abstract
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
$V$
.
Generalized Derivations and Rota-Baxter Operators of $$\varvec{n}$$-ary Hom-Nambu Superalgebras
