arXiv subjects

Cristian Ortiz

Publications and source records attributed to Cristian Ortiz.

16 recordsLinked to original sources

arXiv2024

Vector fields and derivations on differentiable stacks

We introduce and study module structures on both the dgla of multiplicative vector fields and the graded algebra of functions on Lie groupoids. We show that there is an associated structure of a graded Lie-Rinehart algebra on the vector fields of a differentiable stack over its smooth functions that is Morita invariant in an appropriate sense. Furthermore, we show that associated Van-Est type maps are compatible with those module structures. We also present several examples.

math.DG↗

arXiv2024

On the cohomology of differentiable stacks

Morita equivalence classes of Lie groupoids serve as models for differentiable stacks, which are higher spaces in differential geometry, generalizing manifolds and orbifolds. Representations up to homotopy of Lie groupoids provide a higher analog of classical representations and play a significant role in Poisson geometry. In this paper, we prove that the cohomology with coefficients in a representation up to homotopy is a Morita invariant, and therefore an invariant of the underlying stack. This result was inspired by the 2-term case, previously developed by del Hoyo and Ortiz, and it relies on the simplicial approach to representations up to homotopy, recently introduced by del Hoyo and Trentinaglia. As a subsidiary result, we include a proof of the cohomological descent for higher Lie groupoids.

math.DG↗

arXiv2024

Morse theory on Lie groupoids

In this paper we introduce Morse Lie groupoid morphisms and study their main properties. We show that this notion is Morita invariant which gives rise to a well defined notion of Morse function on differentiable stacks. We show a groupoid version of the Morse lemma which is used to describe the topological behavior of the critical subgroupoid levels of a Morse Lie groupoid morphism around its nondegenerate critical orbits. We also prove Morse type inequalities for certain separated differentiable stacks and construct a Morse double complex whose total cohomology is isomorphic to the Bott-Shulman-Stasheff cohomology of the underlying Lie groupoid. We provide several examples and applications.

math.DG↗

arXiv2023

The Chern-Weil-Lecomte characteristic map for $L_{\infty}$-algebras

In this paper we extend the Chern-Weil-Lecomte characteristic map to the setting of $L_{\infty}$-algebras. In this general framework, characteristic classes of $L_{\infty}$-algebra extensions are defined by means of the Chern-Weil-Lecomte map which takes values in the cohomology of an $L_{\infty}$-algebra with coefficients in a representation up to homotopy. This general set up allows us to recover several known cohomology classes in a unified manner, including: the characteristic class of a Lie 2-algebra, the Ševera class of an exact Courant algebroid and the curvature 3-form of a gerbe with connective structure. We conclude by introducing a Chern-Weil map for principal 2-bundles over Lie groupoids.

math.DG↗

arXiv2023

Complex Dirac structures with constant real index on flag manifolds

In this paper we describe all invariant complex Dirac structures with constant real index on a maximal flag manifold in terms of the roots of the Lie algebra which defines the flag manifold. We also completely classify these structures under the action of $B$-transformations.

math.DG↗

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

Morita equivalences of vector bundles

We study vector bundles over Lie groupoids, known as VB-groupoids, and their induced geometric objects over differentiable stacks. We establish a fundamental theorem that characterizes VB-Morita maps in terms of fiber and basic data, and use it to prove the Morita invariance of VB-cohomology, with implications to deformation cohomology of Lie groupoids and of classic geometries. We discuss applications of our theory to Poisson geometry, providing a new insight over Marsden-Weinstein reduction and the integration of Dirac structures. We conclude by proving that the derived category of VB-groupoids is a Morita invariant, which leads to a notion of VB-stacks, and solves (an instance of) an open question on representations up to homotopy.

math.DG↗

arXiv2017

Obstructions to the integrability of VB-algebroids

VB-groupoids can be thought of as vector bundle objects in the category of Lie groupoids. Just as Lie algebroids are the infinitesimal counterparts of Lie groupoids, VB-algebroids correspond to the infinitesimal version of VB-groupoids. In this work we address the problem of the existence of a VB-groupoid admitting a given VB-algebroid as its infinitesimal data. Our main result is an explicit characterization of the obstructions appearing in this integrability problem as the vanishing of the spherical periods of certain cohomology classes. Along the way, we illustrate our result in concrete examples. Finally, as a corollary, we obtain computable obstructions for a $2$-term representation up to homotopy of Lie algebroid to arise as the infinitesimal counterpart of a smooth such representation of a Lie groupoid.

math.DG↗

arXiv2017

On the Lie 2-algebra of sections of an LA-groupoid

In this work we introduce the category of multiplicative sections of an $\la$-groupoid. We prove that this category carries natural strict Lie 2-algebra structures, which are Morita invariant. As applications, we study the algebraic structure underlying multiplicative vector fields on a Lie groupoid and in particular vector fields on differentiable stacks. We also introduce the notion of geometric vector field on the quotient stack of a Lie groupoid, showing that the space of such vector fields is a Lie algebra. We describe the Lie algebra of geometric vector fields in several cases, including classifying stacks, quotient stacks of regular Lie groupoids and in particular orbifolds, and foliation groupoids.

math.DG↗

arXiv2016

Integration of 2-term representations up to homotopy via 2-functors

Given a representation up to homotopy of a Lie algebroid on a 2-term complex of vector bundles, we define the corresponding holonomy as a strict 2-functor from a Weinstein path 2-groupoid to the gauge 2-groupoid of the underlying 2-term complex. We construct a corresponding transformation 2-groupoid and we prove that the 1-truncation of this 2-groupoid is isomorphic to the Weinstein groupoid of the VB-algebroid associated to a representation up to homotopy. As applications, we describe alternative integration schemes for semi-direct products of Lie 2-algebras and string algebras.

math.DG↗

arXiv2016

VB-algebroid morphisms and representations up to homotopy

We show in this paper that the correspondence between $2$-term representations up to homotopy and $\mathcal{VB}$-algebroids, established by Gracia-Saz and Mehta, holds also at the level of morphisms. This correspondence is hence an equivalence of categories. As an application, we study foliations and distributions on a Lie algebroid, that are compatible both with the linear structure and the Lie algebroid structure. In particular, we show how infinitesimal ideal systems in a Lie algebroid $A$ are related with subrepresentations of the adjoint representation of $A$.

math.DG↗

arXiv2012

Multiplicative Dirac structures

In this paper we introduce multiplicative Dirac structures on Lie groupoids, providing a unified framework to study both multiplicative Poisson bivectors (i.e., Poisson group(oid)s) and multiplicative closed 2-forms (e.g., symplectic groupoids). We prove that for every source simply connected Lie groupoid $G$ with Lie algebroid $AG$, there exists a one-to-one correspondence between multiplicative Dirac structures on $G$ and Dirac structures on $AG$, which are compatible with both the linear and algebroid structures of $AG$. We explain in what sense this extends the integration of Lie bialgebroids to Poisson groupoids carried out in \cite{MX2} and the integration of Dirac manifolds of \cite{BCWZ}. We also explain the connection between multiplicative Dirac structures and higher geometric structures such as $\mathcal{LA}$-groupoids and $\mathcal{CA}$-groupoids.

math.DG↗

arXiv2012

Foliated groupoids and their infinitesimal data

In this work, we study Lie groupoids equipped with multiplicative foliations and the corresponding infinitesimal data. We determine the infinitesimal counterpart of a multiplicative foliation in terms of its core and sides together with a partial connection satisfying special properties, giving rise to the concept of IM-foliation on a Lie algebroid. The main result of this paper shows that if $G$ is a source simply connected Lie groupoid with Lie algebroid $A$, then there exists a one-to-one correspondence between multiplicative foliations on $G$ and IM-foliations on the Lie algebroid $A$.

math.DG↗

arXiv2011

B-field transformations of Poisson groupoids

In this work we study B-field transformations of multiplicative Poisson bivectors on a Lie groupoid G. We are concerned with B-fields given by multiplicative closed 2-forms on G. We view Poisson groupoids and their B-field symmetries as special instances of multiplicative Dirac structures. These are geometric structures that unify both multiplicative Poisson bivectors and multiplicative closed 2-forms. This allows us to extend results of Bursztyn and Radko on gauge transformations of symplectic/Poisson groupoids. We also describe B-field symmetries of Poisson groupoids at the infinitesimal level.

math.SG↗

arXiv2009

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.

math.DG↗

arXiv2009

Multiplicative Dirac structures on Lie groups

We study multiplicative Dirac structures on Lie groups. We show that the characteristic foliation of a multiplicative Dirac structure is given by the cosets of a normal Lie subgroup and, whenever this subgroup is closed, the leaf space inherits the structure of a Poisson-Lie group. We also describe multiplicative Dirac structures on Lie groups infinitesimally.

math.SG↗