A note on quadratic Poisson brackets on ${\mathrm{gl}}(n,{\mathbb{R}})$ related to Toda lattices
| Autor: | L. Fehér, B. Juhász |
|---|---|
| Rok vydání: | 2022 |
| Předmět: |
High Energy Physics - Theory
Nonlinear Sciences - Exactly Solvable and Integrable Systems High Energy Physics - Theory (hep-th) FOS: Physical sciences Statistical and Nonlinear Physics Mathematical Physics (math-ph) Exactly Solvable and Integrable Systems (nlin.SI) Mathematics::Symplectic Geometry Mathematical Physics |
| DOI: | 10.48550/arxiv.2204.02077 |
| Popis: | It is well known that the compatible linear and quadratic Poisson brackets of the full symmetric and of the standard open Toda lattices are restrictions of linear and quadratic $r$-matrix Poisson brackets on the associative algebra ${\mathrm{gl}}(n,{\mathbb{R}})$. We here show that the quadratic bracket on ${\mathrm{gl}}(n,{\mathbb{R}})$, corresponding to the $r$-matrix defined by the splitting of ${\mathrm{gl}}(n,{\mathbb{R}})$ into the direct sum of the upper triangular and orthogonal Lie subalgebras, descends by Poisson reduction from a quadratic Poisson structure on the cotangent bundle $T^*{\mathrm{GL}}(n,{\mathbb{R}})$. This complements the interpretation of the linear $r$-matrix bracket as a reduction of the canonical Poisson bracket of the cotangent bundle. Comment: 8 pages, corrected typos in v2 |
| Databáze: | OpenAIRE |
| Externí odkaz: |
|