Papers › Algebras with representable representations
Algebras with representable representations
Xabier García-Martínez, Matsvei Tsishyn, Tim Van der Linden, Corentin Vienne
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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→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 generalised derivations characterise 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 characterise 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 emphasises 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.
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