论文标题
Bialgebras，Manin Triples，古典$ R $ - 久违和前代代数的相干分类结构
Coherent categorical structures for Lie bialgebras, Manin triples, classical $r$-matrices and pre-Lie algebras
论文作者
论文摘要
Bialgebras，Manin Triples，Classical $ r $ - 标准和$ \ Mathcal {O} $ - Lie代数的运营商的广泛应用概念应归功于它们之间的密切关系。然而，这些概念及其对应关系通常被理解为类别之间的对象和地图。为了获得分类洞察力，本文介绍了每个类别的同构概念，统一称为相干同态，以便对象类成为类别，并且类别之间的地图成为函子或类别等价。为此，我们从一个谎言代数的概念开始，由配备有谎言代数内态的谎言代数组成。然后，我们将上述古典概念概括为Lie代数为Endo Lie代数。结果，我们获得了每个类别的相干性内态的概念，然后通过极化过程将其概括为相干同态的概念。相干同态与各种构造之间的对应关系以及前代代数类别兼容。
The broadly applied notions of Lie bialgebras, Manin triples, classical $r$-matrices and $\mathcal{O}$-operators of Lie algebras owe their importance to the close relationship among them. Yet these notions and their correspondences are mostly understood as classes of objects and maps among the classes. To gain categorical insight, this paper introduces, for each of the classes, a notion of homomorphisms, uniformly called coherent homomorphisms, so that the classes of objects become categories and the maps among the classes become functors or category equivalences. For this purpose, we start with the notion of an endo Lie algebra, consisting of a Lie algebra equipped with a Lie algebra endomorphism. We then generalize the above classical notions for Lie algebras to endo Lie algebras. As a result, we obtain the notion of coherent endomorphisms for each of the classes, which then generalizes to the notion of coherent homomorphisms by a polarization process. The coherent homomorphisms are compatible with the correspondences among the various constructions, as well as with the category of pre-Lie algebras.
