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DATE
:
2021-06-24

UNIVERSAL IDENTIFIER
:
http://hdl.handle.net/11093/2701

UNESCO SUBJECT
:
12 Matemáticas ; 1201 Álgebra

DOCUMENT TYPE
:
article

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 ... [+]

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