Topological Lie bialgebra structures and their classification over $ \mathfrak{g}[\![x]\!] $

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.