Group invariance of integrable Pfaffian systems

Abstract

Let S be an integrable Pfaffian system. If it is invariant under a transversally free infinitesimal action of a finite dimensional real Lie algebra g and consequently invariant under the local action of a Lie group G, we show that the vertical variational cohomology of S is equal to the Lie algebra cohomology of g with values in the space of the horizontal cohomology in maximum dimension. This result, besides giving an effective algorithm for the computation of the variational cohomology of an invariant Pfaffian system, provides a method for detecting obstructions to the existence of finite or infinitesimal actions leaving a given system invariant.

On the integrability problem for systems of partial differential equations in one unknown function, II

“…A first systematic investigation on the local equivalence of linear differential systems was initiated by Drach, Picard and Vessiot, whereupon it resulted in the well known Théorie de Picard-Vessiot (cf. [16]).…”

“…On account of the transitivity at the groupoid level, we can apply the Jordan-Hölder integration method as outlined in [16] and [14]. When the linear pseudo-group operates transitively on the equations, the integration process can be achieved by just calling out the simple linear Lie groups.…”