Abstract

This thesis is about the study of Lie groupoids endowed with a multiplicative differential 1-form. The motivation of the present work is to study the geometry of PDEs using the formalism of Lie groupoids and multiplicative forms; as such, ideas from the two theories have to be introduced and explained
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from our point of view before new results can be presented. Therefore the thesis can be naturally split in two halves: the first, consisting of chapters 1, 2 and 3, recall the ideas and methods which are used in the second half, where the majority of original results are presented. It is important to remark that when considering multiplicative structures on Lie groupoids we shall employ two points of view: the one using differential forms and the dual picture with distributions. Chapter 1 provides some preliminaries that are used throughout the thesis. The aim of this chapter is twofold: on the one hand, it introduces concepts which may not be familiar to all readers in a way to ease them into the rest of the thesis, while on the other it provides crucial motivation for this work. In particular, some examples coming from Lie pseudogroups are discussed and a new notion of "generalized pseudogroups" is proposed. As many notions come from the classical theory of Pfaffian systems, it is important to understand their geometry conceptually. Chapters 2 and 3 are devoted to the study of this, starting with the easier-to-handle linear picture of relative connections in Chapter 2, passing to the global description of Chapter 3. In Chapter 4, we move to the theory of multiplicative forms on Lie groupoids and their infinitesimal counterparts on Lie algebroids. While this chapter can be read independently from the rest of the thesis, it presents ideas which are crucial to the understanding of Chapters 5 and 6. The main result of this chapter is the integrability theorem for multiplicative k-form with coefficients, which states that under the usual conditions we can recover a given multiplicative k-form from its infinitesimal data (a k-Spencer operator). Of course, k=1 is the relevant case for the thesis. Chapter 5 and 6 are the core of the thesis in terms of original results. In Chapter 6 everything comes together. The multiplicativity condition for Pfaffian groupoids simplifies the theory developed in Chapter 3, where all the notions here become "Lie theoretic''. In contrast with Chapter 3, integrability results can by applied to ensure that a Pfaffian groupoid can be recovered from its infinitesimal data (its Spencer operator). These are discussed in Chapter 5. Of course, as the linear counterpart of Pfaffian groupoids, they are the natural relative connections on the setting of Lie algebroids: they are compatible with the anchor and the Lie bracket. Chapter 6 also discusses some other results which stem from this thesis, such as the infinitesimal condition that ensures the Frobenius involutivity of the Pfaffian distribution, and integration of Jacobi structures to contact groupoids
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