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

TL;DR

This paper classifies topological Lie bialgebra structures on formal power series Lie algebras over simple Lie algebras, relating them to Manin pairs, trace extensions, and solutions of the classical Yang-Baxter equation.

Contribution

It introduces the notion of topological Manin pairs, relates them to trace extensions, and classifies topological Lie bialgebra structures and their doubles, including explicit results over the complex numbers.

Findings

Classification of topological Lie bialgebra structures via Lagrangian subalgebras.

Identification of only three non-trivial doubles up to equivalence.

Explicit classification over the complex field and solutions to the classical Yang-Baxter equation.

Abstract

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.