Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 22 (2026), 093, 23 pages      arXiv:2602.19885      https://doi.org/10.3842/SIGMA.2026.093
Contribution to the Special Issue on Interactions of Poisson Geometry, Lie Theory and Symmetry in honor of Rui Loja Fernandes

Algebraic Integrability and Minimality of Lie Equations for Transitive, Finite-Dimensional, Non-Commutative Lie Pseudogroups

Alejandro Arenas Tirado a, David Blázquez Sanz b and Guy Casale c
a) Corporación Universitaria Minuto de Dios - UNIMINUTO, Bello, Colombia
b) Departamento de Matemáticas, Facultad de Ciencias, Universidad Nacional de Colombia, Sede Medellín, Colombia
c) Université de Rennes, CNRS, IRMAR-UMR 6625, 35000 Rennes, France

Received February 24, 2026, in final form September 18, 2026; Published online September 29, 2026

Abstract
We provide an algebraic characterization of transitive, finite-dimensional algebraic Lie pseudogroups (or $\mathcal{D}$-groupoids) that are algebraically 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 nontrivial sub-$\mathcal{D}$-groupoids of positive dimension.

Key words: Lie pseudogroup; $\mathcal{D}$-groupoid; differential invariants; Lie connection; Picard-Vessiot group; minimality.

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