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

Alejandro Arenas Tirado, David Blázquez-Sanz, Guy Casale

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.

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

Abstract

We provide an algebraic characterization of transitive, finite-dimensional algebraic Lie pseudogroups (or -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 -groupoid, its algebraic integrability is equivalent to the triviality of the differential Galois group of its -Lie algebra. Furthermore, we investigate the structure of highly non-integrable -groupoids, demonstrating that if the differential Galois group of the linear differential equation of their -Lie algebra is large enough, then they are minimal in the sense that they admit no non-trivial sub--groupoids of positive dimension.
Paper Structure (21 sections, 16 theorems, 73 equations)

This paper contains 21 sections, 16 theorems, 73 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. $\mathcal{D}$-Groupoids or algebraic Lie Pseudogroups
  4. $k$-frame bundles and $k$-jet groupoids
  5. $\mathcal{D}$-Structure of frame bundles
  6. Definition of $\mathcal{D}$-groupoid
  7. $\mathcal{D}_M$-structure of ${\rm Aut}(M)$ and its Kolchin topology
  8. Lie algebra of a $\mathcal{D}$-groupoid
  9. Finite dimensional $\mathcal{D}$-groupoids
  10. Adapted frames
  11. $\mathcal{D}$-Lie algebra and Lie algebroid
  12. Example: projective structures on the complex line
  13. Picard-Vessiot theory
  14. Definition of the Galois group
  15. Galois group for linear connections
  16. ...and 6 more sections

Key Result

Proposition 1

The assignments $\mathcal{G} \leadsto {\rm Inv}(\mathcal{G})$ and $\mathbb F\leadsto {\rm Sym}(\mathbb F)$ are inverse bijective correspondences between the sets of rational subgroupoids of ${\rm Aut}_k(M)$ and $\Gamma_k$ invariant subfields of $\mathbb C(R_kM)$ containing $\mathbb C$.

Theorems & Definitions (48)

  • Example 1
  • Example 2
  • Definition 1
  • Proposition 1
  • Definition 2
  • Proposition 2
  • Proposition 3
  • Definition 3
  • Remark 1
  • Remark 2
  • ...and 38 more