On an infinite commuting ODE system associated to a simple Lie algebra

TL;DR

This paper introduces an infinite family of commuting ODEs linked to simple Lie algebras and connects their tau-functions to well-known integrable hierarchies like KdV and Boussinesq.

Contribution

It constructs a new class of commuting ODEs for each simple Lie algebra and relates their tau-functions to the Drinfeld--Sokolov hierarchy.

Findings

Tau-functions match those of the Drinfeld--Sokolov hierarchy.

Explicit examples for A1 and A2 relate to KdV and Boussinesq hierarchies.

Provides a new perspective on integrable systems associated with Lie algebras.

Abstract

Inspired by a recent work of Dubrovin [7], for each simple Lie algebra $\mathfrak{g}$, we introduce an infinite family of pairwise commuting ODEs and define their $\tau$-functions. We show that these $\tau$-functions can be identified with the $\tau$-functions for the Drinfeld--Sokolov hierarchy of $\mathfrak{g}$-type. Explicit examples for $\mathfrak{g}=A_1$ and $A_2$ are provided, which are connected to the KdV hierarchy and the Boussinesq hierarchy respectively.