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

TL;DR

This paper explores the relationship between first integrals of the full symmetric Toda system and vector fields on the orthogonal group, revealing new representations of invariant functions on Lie algebra duals.

Contribution

It extends the connection between Toda system integrals and vector fields on $SO_n(\R)$, providing a novel representation of invariant functions on Lie algebra duals.

Findings

Established a link between Toda first integrals and vector fields on $SO_n(\R)$

Described a new representation of $B^+(\\\ ext{R})$-invariant functions on $\mathfrak{sl}_n(\R)^*$

Extended the understanding of the Toda system's integrals in geometric terms

Abstract

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 $SO_n(\R)$.