arXiv subjects

Maria Amelia Salazar

Publications and source records attributed to Maria Amelia Salazar.

11 recordsLinked to original sources

arXiv2026

On Relative Cohomology for Lie groupoids and Lie Algebroids

Motivated by our attempt to understand characteristic classes of Lie groupoids and geometric structures, we are brought back to the fundamentals of the cohomology theories of Lie groupoids and algebroids. One element that was missing in the literature was the notion of relative cohomology in this setting. The main aim of this paper is to develop the structural theory of this notion, the relation between the relative cohomology of groupoids and that of algebroids via van Est maps, and to indicate how it can be used to provide an intrinsic definition of characteristic classes.

math.DG↗

arXiv2021

Multiplicative Gray stability

In this paper we prove Gray stability for compact contact groupoids and we use it to prove stability results for deformations of the induced Jacobi bundles.

math.SG↗

arXiv2020

From PDEs to Pfaffian fibrations

We explain how to encode the essential data of a PDE on jet bundle into a more intrinsic object called Pfaffian fibration. We provide motivations to study this new notion and show how prolongations, integrability and linearisations of PDEs generalise to this setting.

math.DG↗

arXiv2019

Contact Dual Pairs

We introduce and study the notion of contact dual pair adopting a line bundle approach to contact and Jacobi geometry. A contact dual pair is a pair of Jacobi morphisms defined on the same contact manifold and satisfying a certain orthogonality condition. Contact groupoids and contact reduction are the main sources of examples. Among other properties, we prove the Characteristic Leaf Correspondence Theorem for contact dual pairs which parallels the analogous result of Weinstein for symplectic dual pairs.

math.DG↗

arXiv2019

Local formulas for multiplicative forms

We provide explicit formulas for integrating multiplicative forms on local Lie groupoids in terms of infinitesimal data. Combined with our previous work [8], which constructs the local Lie groupoid of a Lie algebroid, these formulas produce concrete integrations of several geometric stuctures defined infinitesimally. In particular, we obtain local integrations and non-degenerate realizations of Poisson, Nijenhuis-Poisson, Dirac, and Jacobi structures by local symplectic, symplectic-Nijenhuis, presymplectic, and contact groupoids, respectively.

math.DG↗

arXiv2019

Van Est differentiation and integration

The classical Van Est theory relates the smooth cohomology of Lie groups with the cohomology of the associated Lie algebra, or its relative versions. Some aspects of this theory generalize to Lie groupoids and their Lie algebroids. In this paper, continuing an idea from [18], we revisit the van Est theory using the Perturbation Lemma from homological algebra. Using this technique, we obtain precise results for the van Est differentiation and integrations maps at the level of cochains. Specifically, we construct homotopy inverses to the van Est differentiation maps that are right inverses at the cochain level.

math.DG↗

arXiv2018

On local integration of Lie brackets

We give a direct, explicit and self-contained construction of a local Lie groupoid integrating a given Lie algebroid which only depends on the choice of a spray vector field lifting the underlying anchor map. This construction leads to a complete account of local Lie theory and, in particular, to a finite-dimensional proof of the fact that the category of germs of local Lie groupoids is equivalent to that of Lie algebroids.

math.DG↗

arXiv2017

Contact Isotropic Realisations of Jacobi Manifolds via Spencer Operators

Motivated by the importance of symplectic isotropic realisations in the study of Poisson manifolds, this paper investigates the local and global theory of contact isotropic realisations of Jacobi manifolds, which are those of minimal dimension. These arise naturally when considering multiplicity-free actions in contact geometry, as shown in this paper. The main results concern a classification of these realisations up to a suitable notion of isomorphism, as well as establishing a relation between the existence of symplectic and contact isotropic realisations for Poisson manifolds. The main tool is the classical Spencer operator which is related to Jacobi structures via their associated Lie algebroid, which allows to generalise previous results as well as providing more conceptual proofs for existing ones.

math.SG↗

arXiv2014

Jacobi structures and Spencer operators

This paper explains the fundamental relation between Jacobi structures and the classical Spencer operator coming from the theory of PDEs so as to provide a direct and geometric approach to the integrability of Jacobi structures. It uses recent results on the integrability of Spencer operators and multliplicative forms on Lie groupoids with non-trivial coefficients. In Theorem 1 we show that the Spencer operator associated to a contact groupoid reveals that the base manifold carries a Jacobi structure. Theorem 2 deals with the problem of integrating Jacobi structures to contact groupoids.

math.DG↗

arXiv2013

Pfaffian groupoids

This thesis is about the study of Lie groupoids endowed with a compatible (multiplicative) differential 1-form. The motivation and scope 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 from our point of view (which may not be the same as in the literature!) 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 (equivalent) points of view: the one using differential forms and the dual picture with distributions.

math.DG↗

arXiv2012

Multiplicative Forms and Spencer Operators

Motivated by our attempt to recast Cartan's work on Lie pseudogroups in a more global and modern language, we are brought back to the question of understanding the linearization of multiplicative forms on groupoids and the corresponding integrability problem. From this point of view, the novelty of this paper is that we study forms with coefficients. However, the main contribution of this paper is conceptual: the finding of the relationship between multiplicative forms and Cartan's work, which provides a completely new approach to integrability theorems for multiplicative forms. Back to Cartan, the multiplicative point of view shows that, modulo Lie's functor, the Cartan Pfaffian system (itself a multiplicative form with coefficients!) is the same thing as the classical Spencer operator.

math.DG↗