论文标题
具有代表表示的代数
Algebras with representable representations
论文作者
论文摘要
就像小组动作由小组自动形态学代表一样，代数动作也以派生为代表：直至同构，谎言代数$ b $的分裂代数代数代数$ x $对应于lie代数的代数形式$ b \ b \ b \ b to \ mathit {der x） $ x $的推导。在本文中，我们研究了一个问题是否可以将推导的概念扩展到其他类型的非求解代数上，而不是字段$ \ mathbb {k} $，以使这些广义推论表征$ \ mathbb {K k} $ - 代数 - 代数 - 代数动作。我们证明答案是否定的，只要字段$ \ mathbb {k} $是无限的。实际上，我们证明了一个更强大的结果：所有阿伯利亚行动（通常称为表示或贝克模块）的代表性已经足够了。因此，我们将各种特征范围的谎言代数的多样性与$ 2 $不同，这是唯一的非缔合代数，这是一个非亚伯利亚类别，具有可代表性的代数。这强调了线性内态的Lie代数$ \ Mathfrak {gl}（v）$作为矢量空间$ v $上表示的对象的独特作用。
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 generalised derivations characterise 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 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 $\mathfrak{gl}(V)$ as a representing object for the representations on a vector space $V$.
