Relative algebroids and symmetries of Pfaffian fibrations

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.