Banach Poisson-Lie groups, Lax equations and the AKS theorem in infinite dimensions

TL;DR

This paper extends the theory of Poisson-Lie groups, Lax equations, and the AKS theorem to infinite-dimensional Banach spaces, providing new insights and methods for solving Lax equations in this setting.

Contribution

It develops the theory of Banach Poisson-Lie groups and proves an AKS theorem version for infinite-dimensional Banach algebras, with applications to Lax equations.

Findings

Established a Banach Lie-Poisson space framework.

Proved an AKS theorem for Banach algebras with R-matrices.

Applied results to the semi-infinite Toda lattice.

Abstract

In this paper, we investigate the theory of $R$-brackets, Baxter brackets and Nijenhuis brackets in the Banach setting, in particular in relation with Banach Poisson-Lie groups. The notion of Banach Lie-Poisson space with respect to an arbitrary duality pairing is crucial for the equations of motion to make sense. In the presence of a non-degenerate invariant pairing on a Banach Lie algebra, these equations of motion assume a Lax form. We prove a version of the Adler-Kostant-Symes theorem adapted to $R$-matrices on infinite-dimensional Banach algebras. Applications to the resolution of Lax equations associated to some Banach Manin triples are given. The semi-infinite Toda lattice is also presented as an example of this approach.