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

TL;DR

The paper develops a bridge between the full symmetric Toda system on $Symm_n(\mathbb{R})$ and vector fields on $SO_n(\mathbb{R})$ by linking first integrals to $SO_n$-flows and representing $B^+_n(\mathbb{R})$-invariant functions on $\mathfrak{sl}_n(\mathbb{R})$ as vector fields on $SO_n(\mathbb{R})$. It constructs a Poisson structure on $Symm_n(\mathbb{R})$ via the dual pairing with the Borel and uses projections $M,\bar{M}$ to obtain explicit brackets and Hamiltonians; for $B^+_n(\mathbb{R})$-invariant functions, this yields an Adler-Kostant-Symes–type anti-homomorphism $\{f^s,g^s\}(L)=-\langle L,[\nabla f(L),\nabla g(L)]\rangle$. The main result shows a Lie algebra homomorphism from $B^+_n(\mathbb{R})$-invariant functions to vector fields on $SO_n(\mathbb{R})$ via $\mathcal{T}^{f,\Lambda}(\Psi)=M(\nabla f(\Psi\Lambda\Psi^T))\Psi$, with $[\mathcal{T}^f,\mathcal{T}^g]=\mathcal{T}^{\{f,g\}}$, and the associated Toda flow on $Symm_n$ given by $\dot L=[M(\nabla f(L)),L]$. A detailed $n=4$ example computes explicit $M$-operators for nontrivial invariants, demonstrating noncommutativity with certain integrals while confirming commuting isospectral flows. The work suggests broad generalizations to other Cartan pairs, clarifies the interaction with known Toda symmetries, and points toward geometric interpretations on flag varieties and Kostant-Toda-type systems.

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 $SO_n(\R)$.

Theorems & Definitions (12)

• Remark 2.1
• Remark 2.2
• Proposition 2.3
• Lemma 2.4
• proof
• Remark 2.5
• Remark 2.6
• Proposition 3.1
• Proposition 3.2
• proof