Banach Poisson-Lie groups, Lax equations and the AKS theorem in infinite dimensions
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914179369140224 |
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| author | Goliński, Tomasz Tumpach, Alice Barbora |
| author_facet | Goliński, Tomasz Tumpach, Alice Barbora |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_02107 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Banach Poisson-Lie groups, Lax equations and the AKS theorem in infinite dimensions Goliński, Tomasz Tumpach, Alice Barbora Exactly Solvable and Integrable Systems Mathematical Physics Functional Analysis 37K10, 22E65, 53D17, 22E60, 46T05, 17B38, 58B99 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. |
| title | Banach Poisson-Lie groups, Lax equations and the AKS theorem in infinite dimensions |
| topic | Exactly Solvable and Integrable Systems Mathematical Physics Functional Analysis 37K10, 22E65, 53D17, 22E60, 46T05, 17B38, 58B99 |
| url | https://arxiv.org/abs/2511.02107 |