The main result of the paper is 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 it as a special case of the classification of Manin pairs. The classification of topological doubles states that up to an appropriate 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. For \(F = {\mathbb {C}}\) the classification becomes explicit. Furthermore, this result enables us to classify formal solutions of the classical Yang–Baxter equation.
