On the finite dual of a cocommutative Hopf algebroid. Application to linear differential matrix equations and Picard-Vessiot theory

The main aim of this paper is to give a Hopf algebroid approach to the Picard-Vessiot theory of linear differential matrix equations with coefficients in the polynomial complex algebra. To this end, we introduce a general construction of what we call here the finite dual of a co-commutative (right) Hopf algebroid and then apply this construction to the first Weyl algebra viewed as the universal enveloping Hopf algebroid of the Lie algebroid of all vector fields on the affine complex line. In this way, for a fixed linear differential matrix equation of order >= 1, we are able to recognize the associated algebraic Galois groupoid as a closed subgroupoid of the induced groupoid of the general linear group along the trivial map, and show that is a transitive groupoid (i.e., it has only one type of isotropy algebraic groups). The polynomial coordinate ring of the Galois groupoid turns out to be a Hopf sub-algebroid of the finite dual of the first Weyl algebra and its total isotropy Hopf algebra (the bundle of all isotropy algebraic groups) is recognized as the Picard-Vessiot extension of the polynomial complex algebra for the linear differential equation we started with.