On invariants and equivalence of differential operators under Lie pseudogroups actions

TL;DR

This paper investigates invariants of linear differential operators under algebraic Lie pseudogroup actions, develops normal forms, and addresses the equivalence problem, with applications to local symplectomorphisms.

Contribution

It introduces a method to find invariants and normal forms for differential operators under Lie pseudogroup actions, advancing the classification and equivalence analysis.

Findings

Derived invariants for linear differential operators

Established normal forms using n-invariants principle

Solved the equivalence problem for algebraic Lie pseudogroups

Abstract

In this paper, we study invariants of linear differential operators with respect to algebraic Lie pseudogroups. Then we use these invariants and the principle of n-invariants to get normal forms (or models) of the differential operators and solve the equivalence problem for actions of algebraic Lie pseudogroups. As a running example of application of the methods, we use the pseudogroup of local symplectomorphisms.