Local Integrable Symmetries of Diffieties

TL;DR

This paper studies local integrable symmetries within the framework of diffieties, providing methods to compute these symmetries via PDE systems and exploring their implications for classifying diffieties, especially regarding flatness.

Contribution

It introduces integrable infinitesimal symmetries of diffieties and offers a systematic approach to compute them, advancing the understanding of diffiety classification.

Findings

Reduction of symmetry computation to PDE systems

Examples of linear systems solvable by hand

Insights into diffiety flatness testing

Abstract

In the framework of diffieties, introduced by Vinogradov, we introduce integrable infinitesimal symmetries and show that they define a one parameter pseudogroup of local diffiety morphisms. We prove some preliminary results allowing to reduce the computation of integrable infinitesimal symmetries of a given order to solving a system of partial differential equations.We provide examples for which we can reduce to a linear system that can be solved by hand computation, and investigate some consequences for the local classification of diffiety, with a special interest for testing if a diffiety is flat.