论文标题
拓扑谎言bialgebra结构及其在$ \ mathfrak {g} [\！[x] \！] $上的分类
Topological Lie bialgebra structures and their classification over $ \mathfrak{g}[\![x]\!] $
论文作者
论文摘要
本文专门针对Lie代数$ \ Mathfrak {g} [\！[x] \！] $的拓扑结构分类，其中$ \ mathfrak {g} $是一个有限的少型简单的lie代数，而不是代数封闭的字段$ f $ 0 $ 0 $ 0 $ 0 $ 0 $。 我们介绍了拓扑Manin对$（l，\ Mathfrak {g} [\！[x] \！]）$的概念，并通过将它们关联以跟踪\（f [x] x] \！] \）来提出它们的分类。然后，我们回想起$ \ mathfrak {g} [\！[x] \！] $上的拓扑双重双重双重双重结构的分类，并将后者视为Manin Pairs分类的特殊情况。 拓扑双重的分类指出，直到某些等价概念只有三个非平凡的双打。事实证明，在$ \ mathfrak {g} [\！[x] \！] $上进行的拓扑谎言bialgebra结构与相应双打的某些lagrangian Lie subalgebras进行了两次射击。然后，我们将代数几何数据附加到此类Lagrangian子代数上，并以这种方式获得所有具有非平凡双打的拓扑结构的分类。当$ f = \ mathbb {c} $分类变为明确时。此外，该结果使我们能够对经典杨巴克斯特方程的形式解决方案进行分类。
This paper is devoted to a classification of topological Lie bialgebra structures on the Lie algebra $\mathfrak{g}[\![x]\!]$, where $ \mathfrak{g} $ is a finite-dimensional simple Lie algebra over an algebraically closed field $ F $ of characteristic $ 0 $. We introduce the notion of a topological Manin pair $(L, \mathfrak{g}[\![x]\!])$ and present their classification by relating them to trace extensions of \( F[\![x]\!] \). Then we recall the classification of topological doubles of Lie bialgebra structures on $\mathfrak{g}[\![x]\!]$ and view the latter as a special case of the classification of Manin pairs. The classification of topological doubles states that up to some notion of equivalence there are only three non-trivial doubles. It is proven that topological Lie bialgebra structures on $\mathfrak{g}[\![x]\!]$ are in bijection with certain Lagrangian Lie subalgebras of the corresponding doubles. We then attach algebro-geometric data to such Lagrangian subalgebras and, in this way, obtain a classification of all topological Lie bialgebra structures with non-trivial doubles. When $F = \mathbb{C}$ the classification becomes explicit. Furthermore, this result enables us to classify formal solutions of the classical Yang-Baxter equation.
