# Kostant-Toda lattices and the universal centralizer

## TL;DR

This paper explores the relationship between Kostant-Toda lattices and the integrable system on the universal centralizer of a complex semisimple Lie algebra, revealing a canonical embedding and deeper structural similarities.

## Contribution

It establishes a canonical open embedding of a dense subset of the Kostant-Toda lattice into the universal centralizer's integrable system, highlighting their structural connections.

## Key findings

- A canonical open embedding of a dense subset of the Kostant-Toda lattice into the universal centralizer.
- Qualitative features of the integrable system on the universal centralizer are analyzed.
- The study contextualizes and deepens understanding of the similarities between the two integrable systems.

## Abstract

To each complex semisimple Lie algebra $\mathfrak{g}$ decorated with appropriate data, one may associate two completely integrable systems. One is the well-studied Kostant-Toda lattice, while the second is an integrable system defined on the universal centralizer $\mathcal{Z}_{\mathfrak{g}}$ of $\mathfrak{g}$. These systems are similar in that each exploits and closely reflects the invariant theory of $\mathfrak{g}$, as developed by Chevalley, Kostant, and others. One also has Kostant's description of level sets in the Kostant-Toda lattice, which turns out to suggest deeper similarities between the two integrable systems in question. We study relationships between the two aforementioned integrable systems, partly to understand and contextualize the similarities mentioned above. Our main result is a canonical open embedding of a flow-invariant open dense subset of the Kostant-Toda lattice into $\mathcal{Z}_{\mathfrak{g}}$. Secondary results include some qualitative features of the integrable system on $\mathcal{Z}_{\mathfrak{g}}$.