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Algebras with representable representations

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Algebras with representable representations

Garcia Martinez, Xabier; Tsishyn, M.; Van der Linden, T.; Vienne, C.
 
DATE : 2021-06-24
UNIVERSAL IDENTIFIER : http://hdl.handle.net/11093/2701
UNESCO SUBJECT : 12 Matemáticas ; 1201 Álgebra
DOCUMENT TYPE : article

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→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 ... [+]
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 . [-]

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