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

2025/11/03 by Goliński, Tomasz, Tumpach, Alice Barbora
Mathematics · Physics and Astronomy · #17B38 #22E60 #22E65 #37K10 #46T05 #53D17 #58B99 #Advanced Topics in Algebra #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Mathematics #FOS: Physical sciences #Functional Analysis (math.FA) #Homotopy and Cohomology in Algebraic Topology #Mathematical Physics (math-ph) #Nonlinear Waves and Solitons

paper · doi:10.48550/arxiv.2511.02107

openalex publication_date 2025/11/03 · openalex created_date 2025/11/06 · openalex updated_date 2026/07/28

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.

Citations

• A note on precotangent bundles: the example of Grassmannians
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• Beyond recursion operators
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• Functional analytic background for a theory of infinite-dimensional reductive Lie groups
• Iwasawa decompositions of some infinite-dimensional Lie groups
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• Symplectic leaves in real Banach Lie-Poisson spaces
• Extensions of Banach Lie-Poisson spaces

Related

• Extensions of Banach Lie-Poisson spaces
• On Poisson structures arising from a Lie group action
• Projective Banach Lie bialgebras, the projective Yang-Baxter equation and projective Poisson-Lie groups
• Poisson Brackets in Kontsevich's "Lie World"
• Lie--Poisson pencils related to semisimple Lie algebras: towards\n classification