Isomorphism of Intransitive Linear Lie Equations

TL;DR

This paper demonstrates that formal isomorphisms of intransitive linear Lie equations can be extended locally and introduces intransitive Lie algebras that recover the equations, linking to Cartan's structure functions.

Contribution

It extends formal isomorphisms to neighborhoods and connects intransitive Lie equations with Lie algebras, providing a new structural perspective.

Findings

Formal isomorphisms extend to neighborhoods of transversals.

Intransitive Lie algebra recovers the linear Lie equation.

Connection established between Lie algebra and Cartan's structure functions.

Abstract

We show that formal isomorphism of intransitive linear Lie equations along transversal to the orbits can be extended to neighborhoods of these transversal. In analytic cases, the word formal is dropped from theorems. Also, we associate an intransitive Lie algebra with each intransitive linear Lie equation, and from the intransitive Lie algebra we recover the linear Lie equation, unless of formal isomorphism. The intransitive Lie algebra gives the structure functions introduced by E. Cartan.