Banach Poisson-Lie groups, Lax equations and the AKS theorem in infinite dimensions
Exactly Solvable and Integrable Systems
2025-12-05 v2 Mathematical Physics
Functional Analysis
math.MP
Abstract
In this paper, we investigate the theory of -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 -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.
Cite
@article{arxiv.2511.02107,
title = {Banach Poisson-Lie groups, Lax equations and the AKS theorem in infinite dimensions},
author = {Tomasz Goliński and Alice Barbora Tumpach},
journal= {arXiv preprint arXiv:2511.02107},
year = {2025}
}
Comments
38 pages