Algebras with representable representations

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 Ñ Der pXq from B to the Lie algebra Der pXq 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 K, in such a way that these generalized derivations characterize 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 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 glpV q as a representing object for the representations on a vector space V .