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Alejandro Cabrera

Publications and source records attributed to Alejandro Cabrera.

At least 19 recordsLinked to original sources
arXiv2026

Associative half-densities on symplectic groupoids and quantization

In this paper, we study half-densities enhancing the multiplication map on a symplectic groupoid and which satisfy a suitable associativity condition. This is structurally motivated by the expected complete semiclassical-analytic approximation to a star product for the underlying Poisson manifold. We show the existence and classification of such associative half-densities, and further apply this theory to the understanding of semiclassical factors in Kontsevich's quantization formula. In the particular case of a linear Poisson structure, we recover the factors appearing in the Duflo isomorphism and its Kashiwara-Vergne extensions as a canonical associative enhancement.

math.SG↗
arXiv2026

Geometric differentiation of simplicial manifolds

We provide a complete geometric solution to the problem of differentiating simplicial manifolds, extending classical Lie theory and complementing existing homotopical and formal approaches within a unifying framework. First, we establish a normal form theorem setting a system of compatible tubular neighborhoods. Building on this description, we identify a differentiating ideal in the algebra of cochains, prove that the quotient is semi-free, and interpret it as the Chevalley-Eilenberg algebra of the thus defined higher Lie algebroid. As an application, we introduce a higher version of the van Est map and prove a van Est isomorphism theorem in cohomology, under natural connectivity assumptions. Finally, we identify the algebraic mechanism underlying geometric differentiation as a monoidal refinement of the dual Dold-Kan correspondence, providing a conceptual explanation of the construction and relating it to earlier homotopical and functor-of-points approaches.

math.DG↗
arXiv2024

Riemannian Variational Calculus: Optimal Trajectories Under Inertia, Gravity, and Drag Effects

Robotic motion optimization often focuses on task-specific solutions, overlooking fundamental motion principles. Building on Riemannian geometry and the calculus of variations (often appearing as indirect methods of optimal control), we derive an optimal control equation that expresses general forces as functions of configuration and velocity, revealing how inertia, gravity, and drag shape optimal trajectories. Our analysis identifies three key effects: (i) curvature effects of inertia manifold, (ii) curvature effects of potential field, and (iii) shortening effects from resistive force. We validate our approach on a two-link manipulator and a UR5, demonstrating a unified geometric framework for understanding optimal trajectories beyond geodesic-based planning.

cs.RO↗
arXiv2024

Approximating Symplectic Realizations: A General Framework for the Construction of Poisson Integrators

While the construction of symplectic integrators for Hamiltonian dynamics is well understood, an analogous general theory for Poisson integrators is still lacking. The main challenge lies in overcoming the singular and non-linear geometric behavior of Poisson structures, such as the presence of symplectic leaves with varying dimensions. In this paper, we propose a general approach for the construction of geometric integrators on any Poisson manifold based on independent geometric and dynamic sources of approximation. The novel geometric approximation is obtained by adapting structural results about symplectic realizations of general Poisson manifolds. We also provide an error analysis for the resulting methods and illustrative applications.

math.NA↗
arXiv2024

Reversible out-of-plane to in-plane magnetic transition by electrical and thermal cycling in Ni$_{90}$Fe$_{10}$/BaTiO$_3$(001)

The study investigates the manipulation of the magnetic anisotropy in a thick (1 $\mu$m) Ni$_{90}$Fe$_{10}$ layer electrodeposited on a ferroelectric BaTiO$_3$(001) substrate, using a combination of Magneto-optical Kerr Effect, Photoemission Electron Microscopy with X-ray circular magnetic dichroism and X-ray diffraction. In the as-grown state, the system shows weak perpendicular magnetic anisotropy and characteristic stripe domains. Upon out-of-plane electrical poling of the BaTiO$_3$ substrate, the magnetic anisotropy switches to in-plane with a strong uniaxial behavior. This change is ascribed to the magnetoelastic effect due to the switching of the BaTiO$_3$ ferroelectric [001] axis into the sample plane, as evidenced by XRD. The strong mechanical interaction with the thick Ni$_{90}$Fe$_{10}$ overlayer prevents the full inversion of the substrate. The perpendicular magnetic anisotropy can be recovered by a mild thermal annealing above the BaTiO$_3$ tetrahedral to cubic phase transition and can be cycled by repeated electrical poling/thermal annealing. This method opens the path to a reversible control of the magnetic anisotropy in Ni$_{90}$Fe$_{10}$/BaTiO$_3$ heterostructures from perpendicular to in-plane.

cond-mat.mtrl-sci↗
arXiv2023

Optimal Control of Robotic Systems and Biased Riemannian Splines

In this paper, we study mechanical optimal control problems on a given Riemannian manifold $(Q,g)$ in which the cost is defined by a general cometric $\tilde{g}$. This investigation is motivated by our studies in robotics, in which we observed that the mathematically natural choice of cometric $\tilde{g} = g^{*}$ -- the dual of $g$ -- does not always capture the true cost of the motion. We then, first, discuss how to encode the system's torque-based actuators configuration into a cometric $\tilde{g}$. Second, we provide and prove our main theorem, which characterizes the optimal solutions of the problem associated to general triples $(Q, g, \tilde{g})$ in terms of a 4th order differential equation. We also identify a tensor appearing in this equation as the geometric source of "biasing" of the solutions away from ordinary Riemannian splines and geodesics for $(Q, g)$. Finally, we provide illustrative examples and practical demonstration of the biased splines as providing the true optimizers in a concrete robotics system.

math.OC↗
arXiv2022

Dimensional reduction of Courant sigma models and Lie theory of Poisson groupoids

We show that the 2d Poisson Sigma Model on a Poisson groupoid arises as an effective theory of the 3d Courant Sigma Model associated to the double of the underlying Lie bialgebroid. This field-theoretic result follows from a Lie-theoretic one involving a coisotropic reduction of the odd cotangent bundle by a generalized space of algebroid paths. We also provide several examples, including the case of symplectic groupoids in which we relate the symplectic realization construction of Crainic-Marcut to a particular gauge fixing of the 3d theory.

math-ph↗
arXiv2021

Poisson double structures

We introduce Poisson double algebroids, and the equivalent concept of double Lie bialgebroid, which arise as second-order infinitesimal counterparts of Poisson double groupoids. We develop their underlying Lie theory, showing how these objects are related by differentiation and integration. We use these results to revisit Lie 2-bialgebras by means of Poisson double structures.

math.SG↗
arXiv2020

Generating functions for local symplectic groupoids and non-perturbative semiclassical quantization

This paper contains three results about generating functions for Lie-theoretic integration of Poisson brackets and their relation to quantization. In the first, we show how to construct a generating function associated to the germ of any local symplectic groupoid and we provide an explicit (smooth, non-formal) universal formula $S_\pi$ for integrating any Poisson structure $\pi$ on a coordinate space. The second result involves the relation to semiclassical quantization. We show that the formal Taylor expansion of $S_{t\pi}$ around $t=0$ yields an extract of Kontsevich's star product formula based on tree-graphs, recovering the formal family introduced by Cattaneo, Dherin and Felder in [6]. The third result involves the relation to semiclassical aspects of the Poisson Sigma model. We show that $S_\pi$ can be obtained by non-perturbative functional methods, evaluating a certain functional on families of solutions of a PDE on a disk, for which we show existence and classification.

math.SG↗
arXiv2020

Quotients of multiplicative forms and Poisson reduction

In this paper we study quotients of Lie algebroids and groupoids endowed with compatible differential forms. We identify Lie theoretic conditions under which such forms become basic and characterize the induced forms on the quotients. We apply these results to describe generalized quotient and reduction processes for (twisted) Poisson and Dirac structures, as well as to their integration by (twisted, pre-)symplectic groupoids. In particular, we recover and generalize several known results concerning Poisson reduction.

math.DG↗
arXiv2018

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↗
arXiv2018

Discrete dynamics and differentiable stacks

In this paper we relate the study of actions of discrete groups over connected manifolds to that of their orbit spaces seen as differentiable stacks. We show that the orbit stack of a discrete dynamical system on a simply connected manifold encodes the dynamics up to conjugation and inversion. We also prove a generalization of this result for arbitrary discrete groups and non-simply connected manifolds, and relate it to the covering theory of stacks. As applications, we obtain a geometric version of Rieffel's theorem on irrational rotations of the circle, we compute the stack-theoretic fundamental group of hyperbolic toral automorphisms, and we revisit the classification of lens spaces.

math.DS↗
arXiv2017

About simple variational splines from the Hamiltonian viewpoint

In this paper, we study simple splines on a Riemannian manifold $Q$ 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 curve's 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 $Q$ is the $2$-sphere. We also elaborate on possible applications, including landmark cometrics in computational anatomy.

math.SG↗
arXiv2017

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↗
arXiv2016

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.

math.DG↗
arXiv2015

Dirac Geometry of the Holonomy Fibration

In this paper, we solve the problem of giving a gauge-theoretic description of the natural Dirac structure on a Lie Group which plays a prominent role in the theory of D- branes for the Wess-Zumino-Witten model as well as the theory of quasi-Hamiltonian spaces. We describe the structure as an infinite-dimensional reduction of the space of connections over the circle. Our insight is that the formal Poisson structure on the space of connections is not an actual Poisson structure, but is itself a Dirac structure, due to the fact that it is defined by an unbounded operator. We also develop general tools for reducing Courant algebroids and morphisms between them, allowing us to give a precise correspondence between Hamiltonian loop group spaces and quasi- Hamiltonian spaces.

math.SG↗
arXiv2014

Vector bundles over Lie groupoids and algebroids

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.

math.DG↗