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

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.

arXiv comment38 pages

References in corpus (7)

• Banach Lie-Poisson spaces and reduction
• Integration and geometrization of Rota-Baxter Lie algebras
• Nijenhuis Geometry
• Almost Lie structures on an anchored Banach bundle
• Hierarchy of Hamilton equations on Banach Lie-Poisson spaces related to restricted Grassmannian
• Queer Poisson brackets
• Banach Poisson-Lie groups and Bruhat-Poisson structure of the restricted Grassmannian