论文标题
$ {\ mathrm {gl}}}二次泊松支架的注释（n，{\ mathbb {r}}）$与Toda Lattices有关
A note on quadratic Poisson brackets on ${\mathrm{gl}}(n,{\mathbb{R}})$ related to Toda lattices
论文作者
论文摘要
众所周知，完整对称的兼容的线性和二次泊松支架以及标准的开放式TODA格子是在关联代数上的线性和二次$ r $ -r $ -r $ -Matrix poisson支架的限制。我们在这里表明$ {\ MathRm {gl}}（n，{\ Mathbb {r}}）$上的二次支架，与$ r $ -matrix相对应，通过$ {\ mathrm {gl}}}（n，n，n，{\ mathbbbb {\ rathbb {r}}}和undian atir int trol trot and ables and ablen或lint trot trot and ables和Like subalgebras，从泊松载体中降低了poisson从二次泊松结构上降低，cotangent束$ t^*{\ mathrm {gl}}}（n，n，{\ mathbb {r}}}）$。这补充了对线性$ r $ -matrix支架的解释，以减少cotangent捆绑包的规范泊松支架。
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.
