Topological Manin pairs and $(n,s)$-type series

TL;DR

This paper classifies topological Manin pairs and $(n,s)$-type series related to Lie bialgebra structures, providing a framework for understanding solutions to the generalized Yang-Baxter equation and their applications in integrable systems.

Contribution

It introduces the concept of $(n,s)$-type series, establishes their correspondence with Lagrangian subspaces and quasi-Lie bialgebra structures, and classifies these structures up to twisting and coordinate changes.

Findings

Lagrangian subspaces correspond to skew-symmetric $(n,s)$-type series.

Classification of quasi-Lie bialgebra structures via Manin pairs.

$(n,s)$-type series solve the generalized Yang-Baxter equation.

Abstract

Lie subalgebras of $ L = \mathfrak{g}(\!(x)\!) \times \mathfrak{g}[x]/x^n\mathfrak{g}[x] $, complementary to the diagonal embedding $\Delta$ 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 $\Delta$ 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.