标题： 全对称Toda系统和群上的向量场$SO_n(\R)$ 显示英文标题

标题： Full symmetric Toda system and vector fields on the group $SO_n(\R)$

摘要： 在本文中，我们讨论给出全对称Toda系统（一个重要的在迹为零实对称矩阵空间上的哈密顿系统）首次积分的函数与正交矩阵群上的向量场之间的关系：已知该系统等价于正交群上的常微分方程，我们将这一观察进一步扩展到其首次积分。 作为副产品，我们描述了李代数 $B^+(\R)$-不变函数在李代数 $\mathfrak{sl}_n(\R)$对偶空间上的表示（在规范泊松结构下）通过 $SO_n(\R)$上的向量场。 显示英文摘要

摘要： In this paper we discuss the relation between the functions that give first integrals of full symmetric Toda system (an important Hamilton system on the space of traceless real symmetric matrices) and the vector fields on the group of orthogonal matrices: it is known that this system is equivalent to an ordinary differential equation on the orthogonal group, and we extend this observation further to its first integrals. As a by-product we describe a representation of the Lie algebra of $B^+(\R)$-invariant functions on the dual space of Lie algebra $\mathfrak{sl}_n(\R)$ (under the canonical Poisson structure) by vector fields on $SO_n(\R)$.