Showing 1–19 of 19 results for author: Valencia, F

Stratified vector fields on orbit spaces

Authors: Mateus de Melo, Juan Sebastian Herrera-Carmona, Fabricio Valencia

Abstract: Using Morita type stratifications, we establish a one-to-one correspondence between geometric vector fields on a separated differentiable stack and stratified vector fields on its orbit space. This correspondence enables us to derive a stacky version of the generalized Gauss lemma and to prove a smooth version of Palais' covering isotopy theorem for a class of proper Lie groupoids, thereby extendi… ▽ More Using Morita type stratifications, we establish a one-to-one correspondence between geometric vector fields on a separated differentiable stack and stratified vector fields on its orbit space. This correspondence enables us to derive a stacky version of the generalized Gauss lemma and to prove a smooth version of Palais' covering isotopy theorem for a class of proper Lie groupoids, thereby extending the classical result for proper Lie group actions. △ Less

Submitted 4 May, 2026; originally announced May 2026.

Comments: 16 pages

Lyapunov 1-forms on orbifolds

Authors: Fabricio Valencia

Abstract: We introduce and analyze a notion of smooth Lyapunov 1-form for flows generated by vector fields on orbifolds. Using asymptotic cycles and chain-recurrent sets, we establish topological conditions that guarantee the existence of a Lyapunov 1-form, lying in a prescribed cohomology class, for a given vector field on a compact orbifold. We introduce and analyze a notion of smooth Lyapunov 1-form for flows generated by vector fields on orbifolds. Using asymptotic cycles and chain-recurrent sets, we establish topological conditions that guarantee the existence of a Lyapunov 1-form, lying in a prescribed cohomology class, for a given vector field on a compact orbifold. △ Less

Submitted 1 December, 2025; originally announced December 2025.

Comments: 18 pages. Preliminary version. Comments are very welcome

MSC Class: 22A22; 57R18; 37B25; 37C99

Basic sections of LA-groupoids

Authors: Antonio Maglio, Fabricio Valencia

Abstract: We define the notion of basic section of an LA-groupoid whose core-anchor map is injective. Such a notion turns out to be Morita invariant, so that it provides a simpler model for the sections of the stacky Lie algebroids presented by such LA-groupoids, yet equivalent to the well-known model provided by their multiplicative sections. We define the notion of basic section of an LA-groupoid whose core-anchor map is injective. Such a notion turns out to be Morita invariant, so that it provides a simpler model for the sections of the stacky Lie algebroids presented by such LA-groupoids, yet equivalent to the well-known model provided by their multiplicative sections. △ Less

Submitted 6 November, 2025; originally announced November 2025.

Comments: 13 pages. Comments are welcome

MSC Class: 22A22; 58A30

Lecture notes on the symplectic geometry of graded manifolds and higher Lie groupoids

Authors: Miquel Cueca, Antonio Maglio, Fabricio Valencia

Abstract: In this work, we study symplectic structures on graded manifolds and their global counterparts, higher Lie groupoids. We begin by introducing the concept of graded manifold, starting with the degree 1 case, and translating key geometric structures into classical differential geometry terms. We then extend our discussion to the degree 2 case, presenting several illustrative examples with a particul… ▽ More In this work, we study symplectic structures on graded manifolds and their global counterparts, higher Lie groupoids. We begin by introducing the concept of graded manifold, starting with the degree 1 case, and translating key geometric structures into classical differential geometry terms. We then extend our discussion to the degree 2 case, presenting several illustrative examples with a particular emphasis on equivariant cohomology and Lie bialgebroids. Next, we define symplectic Q-manifolds and their Lagrangian Q-submanifolds, introducing a graded analogue of Weinstein's tubular neighborhood theorem and applying it to the study of deformations of these submanifolds. Shifting focus, we turn to higher Lie groupoids and the shifted symplectic structures introduced by Getzler. We examine their Morita invariance and provide several examples drawn from the literature. Finally, we introduce shifted Lagrangian structures and explore their connections to moment maps and symplectic reduction procedures. Throughout these notes, we illustrate the key constructions and results with concrete examples, highlighting their applications in mathematics and physics. These lecture notes are based on two mini-courses delivered by the first author at Geometry in Algebra and Algebra in Geometry VII (2023) in Belo Horizonte, Brazil, and at the INdAM Intensive Period: Poisson Geometry and Mathematical Physics (2024) in Napoli, Italy. △ Less

Submitted 1 February, 2026; v1 submitted 10 October, 2025; originally announced October 2025.

Comments: 88 pages, V2: modifications from referee, V3: typos corrected

Hamiltonian actions on 0-shifted cosymplectic groupoids

Authors: Daniel López Garcia, Fabricio Valencia

Abstract: We introduce the notion of 0-shifted cosymplectic structure on differentiable stacks and develop a corresponding moment map theory for Hamiltonian cosymplectic actions. We present a reduction procedure, establish a version of the Kirwan convexity theorem, and obtain examples of Morse-Bott Lie groupoid morphisms. We introduce the notion of 0-shifted cosymplectic structure on differentiable stacks and develop a corresponding moment map theory for Hamiltonian cosymplectic actions. We present a reduction procedure, establish a version of the Kirwan convexity theorem, and obtain examples of Morse-Bott Lie groupoid morphisms. △ Less

Submitted 4 March, 2026; v1 submitted 24 August, 2025; originally announced August 2025.

Comments: 12 pages, 1 figure. Comments are very welcome!

MSC Class: 53D20; 57R30; 58H05

Topology of singular foliations of closed 1-forms on orbifolds

Authors: Daniel Lopez Garcia, Fabricio Valencia

Abstract: We study the topological properties of the leaves of the singular foliation induced by a closed 1-form of Morse type on a compact orbifold. In particular, we establish criteria that characterize when all such leaves are compact, when they are non-compact, and how both types may coexist. As an application, we extend to the orbifold setting a celebrated result of Calabi, which provides a purely topo… ▽ More We study the topological properties of the leaves of the singular foliation induced by a closed 1-form of Morse type on a compact orbifold. In particular, we establish criteria that characterize when all such leaves are compact, when they are non-compact, and how both types may coexist. As an application, we extend to the orbifold setting a celebrated result of Calabi, which provides a purely topological characterization of intrinsically closed harmonic 1-forms of Morse type. △ Less

Submitted 3 April, 2026; v1 submitted 13 July, 2025; originally announced July 2025.

Comments: 21 pages, 10 figures/pictures. Comments are very welcome!

MSC Class: 22A22; 57R18; 57R70

Consensus in Models for Opinion Dynamics with Generalized-Bias

Authors: Juan Paz, Camilo Rocha, Luis Tobòn, Frank Valencia

Abstract: Interest is growing in social learning models where users share opinions and adjust their beliefs in response to others. This paper introduces generalized-bias opinion models, an extension of the DeGroot model, that captures a broader range of cognitive biases. These models can capture, among others, dynamic (changing) influences as well as ingroup favoritism and out-group hostility, a bias where… ▽ More Interest is growing in social learning models where users share opinions and adjust their beliefs in response to others. This paper introduces generalized-bias opinion models, an extension of the DeGroot model, that captures a broader range of cognitive biases. These models can capture, among others, dynamic (changing) influences as well as ingroup favoritism and out-group hostility, a bias where agents may react differently to opinions from members of their own group compared to those from outside. The reactions are formalized as arbitrary functions that depend, not only on opinion difference, but also on the particular opinions of the individuals interacting. Under certain reasonable conditions, all agents (despite their biases) will converge to a consensus if the influence graph is strongly connected, as in the original DeGroot model. The proposed approach combines different biases, providing deeper insights into the mechanics of opinion dynamics and influence within social networks. △ Less

Submitted 16 September, 2024; originally announced September 2024.

Double extension of flat pseudo-Riemannian $F$-Lie algebras

Authors: Alexander Torres-Gomez, Fabricio Valencia

Abstract: We define the concept of a flat pseudo-Riemannian $F$-Lie algebra and construct its corresponding double extension. This algebraic structure can be interpreted as the infinitesimal analogue of a Frobenius Lie group devoid of Euler vector fields. We show that the double extension provides a framework for generating all weakly flat Lorentzian non-abelian bi-nilpotent $F$-Lie algebras possessing one… ▽ More We define the concept of a flat pseudo-Riemannian $F$-Lie algebra and construct its corresponding double extension. This algebraic structure can be interpreted as the infinitesimal analogue of a Frobenius Lie group devoid of Euler vector fields. We show that the double extension provides a framework for generating all weakly flat Lorentzian non-abelian bi-nilpotent $F$-Lie algebras possessing one dimensional light-cone subspaces. A similar result can be established for nilpotent Lie algebras equipped with flat scalar products of signature $(2,n-2)$ where $n\geq 4$. Furthermore, we use this technique to construct Poisson algebras exhibiting compatibility with flat scalar products. △ Less

Submitted 28 November, 2024; v1 submitted 3 May, 2024; originally announced May 2024.

Comments: 21 pages. Final version accepted for publication in Journal of Algebra

MSC Class: 53D45; 16E40; 17A30; 22F30; 17B63

On the monodromy action for $f(x,y)=g(x)+h(y)$

Authors: Daniel López Garcia, Fabricio Valencia

Abstract: We solve the problem of determining under which conditions the monodromy of a vanishing cycle generates the whole homology of a regular fiber for a polynomial $f(x,y)=g(x)+h(y)$ where $h$ and $g$ are polynomials with real coefficients having real critical points and satisfying $deg(g)\cdot deg(h)\leq 2500$ and $\gcd(deg(g),deg(h))\leq 2$. As an application, under the same assumptions we solve the… ▽ More We solve the problem of determining under which conditions the monodromy of a vanishing cycle generates the whole homology of a regular fiber for a polynomial $f(x,y)=g(x)+h(y)$ where $h$ and $g$ are polynomials with real coefficients having real critical points and satisfying $deg(g)\cdot deg(h)\leq 2500$ and $\gcd(deg(g),deg(h))\leq 2$. As an application, under the same assumptions we solve the tangential-center problem which is known to be linked to the weak 16th Hilbert problem. △ Less

Submitted 30 September, 2024; v1 submitted 9 November, 2023; originally announced November 2023.

Comments: 11 pages. Final version accepted for publication in Proceedings of the American Mathematical Society

MSC Class: 14D05; 32S40; 34M35

Novikov type inequalities for orbifolds

Authors: Fabricio Valencia

Abstract: We propose a natural extension of the Novikov numbers for the basic cohomology class of a closed basic $1$-form on a proper Lie groupoid of finitely generated type. As an application, we prove corresponding Novikov inequalities for compact orbifolds. We propose a natural extension of the Novikov numbers for the basic cohomology class of a closed basic $1$-form on a proper Lie groupoid of finitely generated type. As an application, we prove corresponding Novikov inequalities for compact orbifolds. △ Less

Submitted 15 May, 2025; v1 submitted 9 June, 2023; originally announced June 2023.

Comments: 25 pages. Final version accepted for publication in Mathematische Annalen

MSC Class: 22A22; 57R18; 57R70

Counting and Computing Join-Endomorphisms in Lattices (Revisited)

Authors: Carlos Pinzón, Santiago Quintero, Sergio Ramírez, Camilo Rueda, Frank Valencia

Abstract: Structures involving a lattice and join-endomorphisms on it are ubiquitous in computer science. We study the cardinality of the set $\mathcal{E}(L)$ of all join-endomorphisms of a given finite lattice $L$. In particular, we show for $\mathbf{M}_n$, the discrete order of $n$ elements extended with top and bottom, $| \mathcal{E}(\mathbf{M}_n) | =n!\mathcal{L}_n(-1)+(n+1)^2$ where $\mathcal{L}_n(x)$… ▽ More Structures involving a lattice and join-endomorphisms on it are ubiquitous in computer science. We study the cardinality of the set $\mathcal{E}(L)$ of all join-endomorphisms of a given finite lattice $L$. In particular, we show for $\mathbf{M}_n$, the discrete order of $n$ elements extended with top and bottom, $| \mathcal{E}(\mathbf{M}_n) | =n!\mathcal{L}_n(-1)+(n+1)^2$ where $\mathcal{L}_n(x)$ is the Laguerre polynomial of degree $n$. We also study the following problem: Given a lattice $L$ of size $n$ and a set $S\subseteq \mathcal{E}(L)$ of size $m$, find the greatest lower bound ${\large\sqcap}_{\mathcal{E}(L)} S$. The join-endomorphism ${\large\sqcap}_{\mathcal{E}(L)} S$ has meaningful interpretations in epistemic logic, distributed systems, and Aumann structures. We show that this problem can be solved with worst-case time complexity in $O(mn)$ for distributive lattices and $O(mn + n^3)$ for arbitrary lattices. In the particular case of modular lattices, we present an adaptation of the latter algorithm that reduces its average time complexity. We provide theoretical and experimental results to support this enhancement. The complexity is expressed in terms of the basic binary lattice operations performed by the algorithm. △ Less

Submitted 1 November, 2022; originally announced November 2022.

Isometric Lie 2-group actions on Riemannian groupoids

Authors: Juan Sebastian Herrera-Carmona, Fabricio Valencia

Abstract: We study isometric actions of Lie $2$-groups on Riemannian groupoids by exhibiting some of their immediate properties and implications. Firstly, we prove an existence result which allows both to obtain 2-equivariant versions of the Slice Theorem and the Equivariant Tubular Neighborhood Theorem and to construct bi-invariant groupoid metrics on compact Lie $2$-groups. We provide natural examples, tr… ▽ More We study isometric actions of Lie $2$-groups on Riemannian groupoids by exhibiting some of their immediate properties and implications. Firstly, we prove an existence result which allows both to obtain 2-equivariant versions of the Slice Theorem and the Equivariant Tubular Neighborhood Theorem and to construct bi-invariant groupoid metrics on compact Lie $2$-groups. We provide natural examples, transfer some classical constructions and explain how this notion of isometric $2$-action yields a way to develop a 2-equivariant Morse theory on Lie groupoids. Secondly, we give an infinitesimal description of an isometric Lie $2$-group action. We define an algebra of transversal infinitesimal isometries associated to any Riemannian $n$-metric on a Lie groupoid which in turn gives rise to a notion of geometric Killing vector field on a quotient Riemannian stack. If our Riemannian stack is separated then we prove that the algebra formed by such geometric Killing vector fields is always finite dimensional. △ Less

Submitted 18 July, 2023; v1 submitted 18 September, 2022; originally announced September 2022.

Comments: 31 pages. Substantial changes have been made. The presentation has been improved, new results and examples have been added. Final version accepted for publication

MSC Class: 22A22; 58H05; 58D19

Morse theory on Lie groupoids

Authors: Cristian Ortiz, Fabricio Valencia

Abstract: 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 nond… ▽ More 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. △ Less

Submitted 2 June, 2024; v1 submitted 15 July, 2022; originally announced July 2022.

Comments: 54 pages. Final version accepted for publication in Mathematische Zeitschrift

MSC Class: 22A22; 58H05; 57R70; 37D15

Invariant generalized almost complex structures on real flag manifolds

Authors: Fabricio Valencia, Carlos Varea

Abstract: We characterize those real flag manifolds that can be endowed with invariant generalized almost complex structures. We show that no $GM_2$-maximal real flag manifolds admit integrable invariant generalized almost complex structures. We give a concrete description of the generalized complex geometry on the maximal real flags of type $B_2$, $G_2$, $A_3$, and $D_l$ with $l\geq 5$, where we prove that… ▽ More We characterize those real flag manifolds that can be endowed with invariant generalized almost complex structures. We show that no $GM_2$-maximal real flag manifolds admit integrable invariant generalized almost complex structures. We give a concrete description of the generalized complex geometry on the maximal real flags of type $B_2$, $G_2$, $A_3$, and $D_l$ with $l\geq 5$, where we prove that the space of invariant generalized almost complex structures under invariant $B$-transformations is homotopy equivalent to a torus and we classify all invariant generalized almost Hermitian structures on them. △ Less

Submitted 18 August, 2022; v1 submitted 16 November, 2021; originally announced November 2021.

Comments: 34 pages. Minor changes have been made. Final version to appear in The Journal of Geometric Analysis

MSC Class: 53D18; 14M15

Left invariant special Kähler structures

Authors: Fabricio Valencia

Abstract: We construct left invariant special Kähler structures on the cotangent bundle of a flat pseudo-Riemannian Lie group. We introduce the twisted cartesian product of two special Kähler Lie algebras according to two linear representations by infinitesimal Kähler transformations. We also exhibit a double extension process of a special Kähler Lie algebra which allows us to get all simply connected speci… ▽ More We construct left invariant special Kähler structures on the cotangent bundle of a flat pseudo-Riemannian Lie group. We introduce the twisted cartesian product of two special Kähler Lie algebras according to two linear representations by infinitesimal Kähler transformations. We also exhibit a double extension process of a special Kähler Lie algebra which allows us to get all simply connected special Kähler Lie groups with bi-invariant symplectic connections. All Lie groups constructed by performing this double extension process can be identified with a subgroup of symplectic (or Kähler) affine transformations of its Lie algebra containing a nontrivial $1$-parameter subgroup formed by central translations. We show a characterization of left invariant flat special Kähler structures using étale Kähler affine representations, exhibit some immediate consequences of the constructions mentioned above, and give several non-trivial examples. △ Less

Submitted 13 December, 2021; v1 submitted 19 May, 2020; originally announced May 2020.

Comments: 26 pages. The title has changed. Final version to appear in Indagationes Mathematicae

MSC Class: 53C25; 53C26; 22E60; 22F30

Invariant generalized complex geometry on maximal flag manifolds and their moduli

Authors: Elizabeth Gasparim, Fabricio Valencia, Carlos Varea

Abstract: We describe moduli spaces of invariant generalized complex structures and moduli spaces of invariant generalized Kähler structures on maximal flag manifolds under $B$-transformations. We give an alternative description of the moduli space of generalized complex structures using pure spinors, and describe a cell decomposition of these moduli spaces induced by the action of the Weyl group. We describe moduli spaces of invariant generalized complex structures and moduli spaces of invariant generalized Kähler structures on maximal flag manifolds under $B$-transformations. We give an alternative description of the moduli space of generalized complex structures using pure spinors, and describe a cell decomposition of these moduli spaces induced by the action of the Weyl group. △ Less

Submitted 15 January, 2021; v1 submitted 18 December, 2019; originally announced December 2019.

Comments: 29 pages. The title has been changed. Final version to appear in Journal of Geometry and Physics

MSC Class: 53D18; 14M15

Notes on flat pseudo-Riemannian manifolds

Authors: Fabricio Valencia

Abstract: In these notes we survey basic concepts of affine geometry and their interaction with Riemannian geometry. We give a characterization of affine manifolds which has as counterpart those pseudo-Riemannian manifolds whose Levi-Civita connection is flat. We show that no connected semisimple Lie group admits a left invariant flat affine connection. We characterize flat pseudo-Riemannian Lie groups. For… ▽ More In these notes we survey basic concepts of affine geometry and their interaction with Riemannian geometry. We give a characterization of affine manifolds which has as counterpart those pseudo-Riemannian manifolds whose Levi-Civita connection is flat. We show that no connected semisimple Lie group admits a left invariant flat affine connection. We characterize flat pseudo-Riemannian Lie groups. For a flat left-invariant pseudo-metric on a Lie group, we show the equivalence between the completeness of the Levi-Civita connection and unimodularity of the group. We emphasize the case of flat left invariant hyperbolic metrics on the cotangent bundle of a simply connected flat affine Lie group. We also discuss Lie groups with bi-invariant pseudo-metrics and the construction of orthogonal Lie algebras. △ Less

Submitted 21 March, 2019; originally announced March 2019.

Comments: 25 pages. To appear in Pro Mathematica 60, (2019)

MSC Class: 52C20; 22E60; 53A15

Infinitesimally Tight Lagrangian Orbits

Authors: Elizabeth Gasparim, Luiz A. B. San Martin, Fabricio Valencia

Abstract: We describe isotropic orbits for the restricted action of a subgroup of a Lie group acting on a symplectic manifold by Hamiltonian symplectomorphisms and admitting an Ad*-equivariant moment map. We obtain examples of Lagrangian orbits of complex flag manifolds, of cotangent bundles of orthogonal Lie groups, and of products of flags. We introduce the notion of infinitesimally tight and study the in… ▽ More We describe isotropic orbits for the restricted action of a subgroup of a Lie group acting on a symplectic manifold by Hamiltonian symplectomorphisms and admitting an Ad*-equivariant moment map. We obtain examples of Lagrangian orbits of complex flag manifolds, of cotangent bundles of orthogonal Lie groups, and of products of flags. We introduce the notion of infinitesimally tight and study the intersection theory of such Lagrangian orbits, giving many examples. △ Less

Submitted 3 August, 2020; v1 submitted 8 March, 2019; originally announced March 2019.

Comments: 22 pages. Final version to appear in Mathematische Zeitschrift

MSC Class: Primary: 53D12; Secondary: 14M15

Flat affine symplectic Lie groups

Authors: Fabricio Valencia

Abstract: We give a new characterization of flat affine manifolds in terms of an action of the Lie algebra of classical infinitesimal affine transformations on the bundle of linear frames. We characterize flat affine symplectic Lie groups using symplectic étale affine representations and as a consequence of this, we show that a flat affine symplectic Lie group with bi-invariant symplectic connection contain… ▽ More We give a new characterization of flat affine manifolds in terms of an action of the Lie algebra of classical infinitesimal affine transformations on the bundle of linear frames. We characterize flat affine symplectic Lie groups using symplectic étale affine representations and as a consequence of this, we show that a flat affine symplectic Lie group with bi-invariant symplectic connection contains a nontrivial one parameter subgroup formed by central translations. We give two methods for constructing flat affine symplectic Lie groups, thus obtaining all those having bi-invariant symplectic connections. We get nontrivial examples of simply connected flat affine symplectic Lie groups in every even dimension. △ Less

Submitted 3 August, 2020; v1 submitted 5 February, 2019; originally announced February 2019.

Comments: 28 pages. Final version to appear in Journal of Lie Theory

MSC Class: Primary: 53D05; 53A15; Secondary: 22E60; 22F30