Relative algebroids and symmetries of Pfaffian fibrations
Wilmer Smilde
TL;DR
The paper reconciles Pfaffian fibrations with relative algebroids by showing that every Pfaffian fibration naturally induces a relative algebroid, and that their prolongations and local solutions correspond. It introduces two symmetry notions—internal and Pfaffian—then develops a framework for Pfaffian groupoid actions by these symmetries, proving invariance of the underlying relative algebroid under such actions. Prolongation theory is extended to groupoids and partial actions, clarifying how symmetries lift to prolongations and how derivations on the relative algebroid behave under symmetry. The work highlights that Pfaffian actions alone do not capture all PDEs with symmetries, motivating a broader Pfaffian-symmetry approach and providing prescriptions for quotients and invariant structures in geometric PDE contexts, including Lie pseudogroup symmetries.
Abstract
Relative algebroids and Pfaffian fibrations are two frameworks recently developed to study geometric structures and PDEs with symmetries, but have structurally different foundations. In this article, we clarify the relation between the two. We show that every Pfaffian fibration canonically induces a relative algebroid, and that their prolongations and local solutions coincide. Moreover, we introduce two notions of symmetries of Pfaffian fibrations, namely internal symmetries and Pfaffian symmetries, and develop the theory for actions of Pfaffian groupoids by internal/Pfaffian symmetries. We show that such actions preserve the underlying relative algebroid in an appropriate sense. Our results apply in particular to partial differential equations with Lie pseudogroup symmetries.