Banach Poisson-Lie groups, Lax equations and the AKS theorem in infinite dimensions
Tomasz Goliński, Alice Barbora Tumpach
TL;DR
<3-5 sentence high-level summary> The paper develops a Banach-analytic framework for integrable systems by extending R-matrix, Baxter, and Nijenhuis theories to Banach Lie algebras and Banach Poisson–Lie groups. It establishes a Banach version of the Adler–Kostant–Symes theorem, links Manin triples with Banach Lie–Poisson spaces, and derives Lax equations on adjoint orbits that are solvable via factorization problems, including Iwasawa decompositions in Schatten classes. Applications include Lax equations for Banach Manin triples and the semi-infinite Toda lattice, illustrating how infinite-dimensional integrable dynamics can be treated with a unified, duality-based geometric approach. The results broaden the toolkit for infinite-dimensional integrable systems, offering both structural (Poisson, bialgebra, Rota-Baxter, Nijenhuis) and concrete dynamical (Lax flows, Toda lattice) insights in Banach settings.
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.