A note on quadratic Poisson brackets on ${\mathrm{gl}}(n,{\mathbb{R}})$ related to Toda lattices

TL;DR

This paper explores quadratic Poisson brackets on the Lie algebra gl(n,R) related to Toda lattices, showing their derivation via Poisson reduction from structures on the cotangent bundle of GL(n,R).

Contribution

It demonstrates that a specific quadratic Poisson bracket on gl(n,R) arises from Poisson reduction of a structure on T*GL(n,R), linking Toda lattice brackets to geometric structures.

Findings

Quadratic brackets on gl(n,R) are restrictions of r-matrix Poisson brackets.

The quadratic bracket corresponds to an r-matrix from a splitting of gl(n,R).

The bracket descends from a Poisson structure on T*GL(n,R).

Abstract

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.