RT Journal Article T1 Algebras with representable representations A1 Garcia Martinez, Xabier A1 Tsishyn, M. A1 Van der Linden, T. A1 Vienne, C. K1 12 Matemáticas K1 1201 Álgebra AB 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→Der(X) from B to the Lie algebra 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 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 gl(V) as a representing object for the representations on a vector space V . PB Proceedings of the Edinburgh Mathematical Society SN 00130915 YR 2021 FD 2021-06-24 LK http://hdl.handle.net/11093/2701 UL http://hdl.handle.net/11093/2701 LA eng NO Proceedings of the Edinburgh Mathematical Society, 64(3): 555-573 (2021) NO Financiado para publicación en acceso aberto: Universidade de Vigo/CISUG NO Ministerio de Economía y Competitividad | Ref. MTM2016-79661-P DS Investigo RD 16-feb-2025