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 $B\to {\mathit {Der}}(X)$ from $B$ to the Lie algebra ${\mathit {Der}}(X)$ 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 ${\mathbb {K}}$, in such a way that these generalized derivations characterize the ${\mathbb {K}}$-algebra actions. We prove that the answer is no, as soon as the field ${\mathbb {K}}$ 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 $\mathfrak {gl}(V)$ as a representing object for the representations on a vector space $V$.

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具有可表示表示的代数

就像群动作由群自同构表示一样，李代数动作由派生表示：直到同构，李代数的分裂扩展$B$通过李代数$X$对应于李代数态射$B\to {\mathit {Der}}(X)$从$B$到李代数${\mathit {Der}}(X)$的推导$X$. 在本文中，我们研究了导数的概念是否可以扩展到域上其他类型的非结合代数的问题${\mathbb {K}}$，以这样一种方式，这些广义推导表征${\mathbb {K}}$-代数动作。我们证明答案是否定的，只要现场${\mathbb {K}}$是无限的。事实上，我们证明了一个更强有力的结果：所有的可表示性已经阿贝尔动作——通常称为陈述要么贝克模块——足以证明这一点。因此，我们在一个无限的特征域上刻画了李代数的多样性，不同于$2$作为唯一一种非结合代数，它是具有可表示表示的非阿贝尔范畴。这强调了线性自同态的李代数所起的独特作用$\mathfrak {gl}(V)$作为向量空间上表示的表示对象$V$.