Symmetries of trivial systems of ODEs of mixed order

TL;DR

This paper investigates the symmetry algebras of trivial mixed-order ODE systems, revealing multiple algebraic structures and their relations to prolongation theories, with specific examples illustrating unexpected algebraic properties.

Contribution

It analyzes symmetry algebras of specific trivial ODE systems, exploring their relation to prolongation methods and providing new examples of algebraic phenomena.

Findings

Multiple symmetry algebra series for given ODE systems

Existence of non-isomorphic algebras with same dimension

Connection between symmetry algebras and prolongation theories

Abstract

We compute symmetry algebras of a system of two equations y^(k)=z^(l)=0, where 2<=k<l. It appears that there are many ways to convert such system of ODEs to an exterior differential system. They lead to different series of finite-dimensional symmetry algebras. For example, for (k,l)=(2,3) we get two non-isomorphic symmetry algebras of the same dimension. We explore how these symmetry algebras are related to both Sternberg prolongation of G-structures and Tanaka prolongation of graded nilpotent Lie algebras. Surprisingly, the case (k,l)=(2,3) provides an example of a linear subalgebra g in gl(5,R) such that the Sternberg prolongations of g and g^t are both of the same dimension, but are non-isomorphic. We also discuss the non-linear case and the link with flag structures on smooth manifolds.