Computing almost commuting bases of ODOs and Gelfand-Dickey hierarchies

TL;DR

This paper develops computational methods to construct bases of almost commuting operators related to Gelfand-Dickey hierarchies, avoiding pseudo-differential operators, and implements these algorithms in SageMath.

Contribution

It introduces a new computational approach and SageMath package for constructing bases of almost commuting operators in the ring of ODOs, simplifying the derivation of GD hierarchies.

Findings

Successful computation of almost commuting bases using the new algorithms.

Implementation of the methods in SageMath package 'dalgebra'.

Explicit examples demonstrating the approach's effectiveness.

Abstract

Almost commuting operators were introduced in 1985 by George Wilson to present generalizations of the Korteweg-de Vries hierarchy, nowadays known as Gelfand-Dickey (GD) hierarchies. In this paper, we review the formal construction of the vector space of almost commuting operators with a given ordinary differential operator (ODO), with the ultimate goal of obtaining a basis by computational routines, using the language of differential polynomials. We use Wilson's results on weighted ODOs to guarantee the solvability of the triangular system that allows to compute the homogeneous almost commuting operator of a given order in the ring of ODOs. As a consequence, the computation of the equations of the GD hierarchies is achieved without using pseudo-differential operators. A new package in SageMath called \texttt{dalgebra} has been designed to perform symbolic calculations in differential domains. The algorithms to calculate the almost commuting basis and the GD hierarchies in the ring of ODOs are implemented in SageMath, and explicit examples are provided.