Alejandro Cabrera

Linear and Multiplicative 2-Forms

We study the relationship between multiplicative 2-forms on Lie groupoids and linear 2-forms on Lie algebroids, which leads to a new approach to the infinitesimal description of multiplicative 2-forms and to the integration of twisted Dirac manifolds.

Vector bundles over Lie groupoids and algebroids

Abstract We study VB-groupoids and VB-algebroids, which are vector bundles in the realm of Lie groupoids and Lie algebroids. Through a suitable reformulation of their definitions, we elucidate the Lie theory relating these objects, i.e., their relation via differentiation and integration. We also show how to extend our techniques to describe the more general Lie theory underlying double Lie algebroids and LA-groupoids.

Van Est isomorphism for homogeneous cochains

VB-groupoids define a special class of Lie groupoids which carry a compatible linear structure. In this paper, we show that their differentiable cohomology admits a refinement by considering the complex of cochains which are k-homogeneous on the linear fiber. Our main result is a Van Est theorem for such cochains. We also work out two applications to the general theory of representations of Lie groupoids and algebroids. The case k=1 yields a Van Est map for representations up to homotopy on 2-term graded vector bundles. Arbitrary k-homogeneous cochains on suitable VB-groupoids lead to a novel Van Est theorem for differential forms on Lie groupoids with values in a representation.

Multisymplectic Geometry and Lie Groupoids

We study higher-degree generalizations of symplectic groupoids, referred to as multisymplectic groupoids. Recalling that Poisson structures may be viewed as infinitesimal counterparts of symplectic groupoids, we describe “higher” versions of Poisson structures by identifying the infinitesimal counterparts of multisymplectic groupoids. Some basic examples and features are discussed.

About simple variational splines from the Hamiltonian viewpoint

In this paper, we study simple splines on a Riemannian manifold

Dirac Geometry of the Holonomy Fibration

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On local integration of Lie brackets

from the point of view of the Pontryagin maximum principle (PMP) in optimal control theory. The control problem consists in finding smooth curves matching two given tangent vectors with the control being the curves acceleration, while minimizing a given cost functional. We focus on cubic splines (quadratic cost function) and on time-minimal splines (constant cost function) under bounded acceleration. We present a general strategy to solve for the optimal hamiltonian within the PMP framework based on splitting the variables by means of a linear connection. We write down the corresponding hamiltonian equations in intrinsic form and study the corresponding hamiltonian dynamics in the case

Formal Symplectic Realizations

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Differentiability of correlations in realistic quantum mechanics

is the

Obstructions to the integrability of VB-algebroids

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