论文标题
拓扑结对和$（n，s）$ - 类型系列
Topological Manin pairs and $(n,s)$-type series
论文作者
论文摘要
为$ l = \ mathfrak {g}（\！（x）\！）\ times \ times \ mathfrak {g} [x]/x^n \ mathfrak {g} [x] $，与对角嵌入$ \ m mathfrak的$ \ mathfrak {与正式的$ r $ r $ r $ r $ r $ r $ r $ r $ lie bialgebra结构进行培养。在这项工作中，我们考虑了$ l $补充$δ$的任意子空间，并将它们与所谓的一系列$ $（N，S）$相关联。 我们证明Lagrangian子空间与偏斜 - 对称$（n，s）$ -Type系列和拓扑Quasi-lie bialgebra结构进行了两者，$ \ mathfrak {g} [\！[x] \！] $。使用Manin对的分类，我们将所有准双子结构分类为扭曲和协调转换。 一系列$（n，s）$的类型求解了广义的杨巴克斯特方程，对应于$ l $的子代理。我们讨论了它们在可集成系统理论中的可能实用性。
Lie subalgebras of $ L = \mathfrak{g}(\!(x)\!) \times \mathfrak{g}[x]/x^n\mathfrak{g}[x] $, complementary to the diagonal embedding $Δ$ of $ \mathfrak{g}[\![x]\!] $ and Lagrangian with respect to some particular form, are in bijection with formal classical $r$-matrices and topological Lie bialgebra structures on the Lie algebra of formal power series $ \mathfrak{g}[\![x]\!] $. In this work we consider arbitrary subspaces of $ L $ complementary to $Δ$ and associate them with so-called series of type $ (n,s) $. We prove that Lagrangian subspaces are in bijection with skew-symmetric $ (n,s) $-type series and topological quasi-Lie bialgebra structures on $ \mathfrak{g}[\![x]\!] $. Using the classificaiton of Manin pairs we classify up to twisting and coordinate transformations all quasi-Lie bialgebra structures. Series of type $ (n,s) $, solving the generalized Yang-Baxter equation, correspond to subalgebras of $L$. We discuss their possible utility in the theory of integrable systems.
