Algebraic integrability and minimality of Lie equations for transitive, finite dimensional, non-commutative pseudogroups

TL;DR

This paper characterizes algebraic integrability of transitive, finite-dimensional Lie pseudogroups using differential Galois theory, linking integrability to the triviality of the Galois group and exploring minimality in non-integrable cases.

Contribution

It introduces an algebraic characterization of integrable Lie pseudogroups via differential Galois groups and analyzes the structure of non-integrable pseudogroups for the first time.

Findings

Algebraic integrability is equivalent to the triviality of the differential Galois group.

Highly non-integrable pseudogroups are minimal, with no non-trivial sub-$$-groupoids.

The approach connects differential Galois theory with the structure of Lie pseudogroups.

Abstract

We provide an algebraic characterization of transitive, finite-dimensional algebraic Lie pseudogroups (or $\mathcal{D}$-groupoids) that are algebraic integrable, that is, isogenous to the action groupoid of an algebraic group action. Our approach is based on the differential Galois theory of rational connections. Under suitable hypotheses on the Lie algebra of the $\mathcal D$-groupoid, its algebraic integrability is equivalent to the triviality of the differential Galois group of its $\mathcal D$-Lie algebra. Furthermore, we investigate the structure of highly non-integrable $\mathcal{D}$-groupoids, demonstrating that if the differential Galois group of the linear differential equation of their $\mathcal D$-Lie algebra is large enough, then they are minimal in the sense that they admit no non-trivial sub-$\mathcal{D}$-groupoids of positive dimension.