Proper Lie Research Articles

Algebraic method for multisensor data fusion.

In this contribution, we use Gaussian posterior probability densities to characterize local estimates from distributed sensors, and assume that they all belong to the Riemannian manifold of Gaussian distributions. Our starting point is to introduce a proper Lie algebraic structure for the Gaussian submanifold with a fixed mean vector, and then the average dissimilarity between the fused density and local posterior densities can be measured by the norm of a Lie algebraic vector. Under Gaussian assumptions, a geodesic projection based algebraic fusion method is proposed to achieve the fused density by taking the norm as the loss. It provides a robust fixed point iterative algorithm for the mean fusion with theoretical convergence, and gives an analytical form for the fused covariance matrix. The effectiveness of the proposed fusion method is illustrated by numerical examples.

Bioconvective flow of bi-viscous Bingham nanofluid subjected to Thompson and Troian slip conditions

This paper describes the bioconvection phenomenon and its significant influence on the thermal features of the flow of bi-viscous Bingham (BVB) nanofluid past a vertically stretching flat surface. The analysis of the impact of convection parameters is considered along with various other forces. Meanwhile, the flow of BVB nanofluid is put through the slip conditions defined by Thompson and Troian for the velocity at the boundary. The flow of BVB nanofluid is modeled using the partial differential equations (PDEs) under the assumptions of thermophoresis and Brownian motion which occur due to the movement of nanoparticles. Along with these forces, the radiation is also considered so that the obtained results are close to the practical scenarios. Thus, using the proper Lie group similarity transformations, the intended mathematical model is converted into ordinary differential equations (ODEs). The resulting equation system is encoded using the RKF-45 technique, and the outcomes are explained using graphs and tables. The solutions found for the model showed that, for higher ranges of the non-Newtonian fluid parameter, the velocity decreases while the heat transferred by the nanofluid increases. The availability of motile density at the surface grows as the Péclet number rises, whereas the Schmidt numbers decline in their respective profiles.

Multiplicative Ehresmann connections

We develop the theory of multiplicative Ehresmann connections for Lie groupoid submersions covering the identity, as well as their infinitesimal counterparts. We construct obstructions to the existence of such connections, and we prove existence for several interesting classes of Lie groupoids and Lie algebroids, including all proper Lie groupoids. We show that many notions from the theory of principal bundle connections have analogues in this general setup, including connections 1-forms, curvature 2-forms, Bianchi identity, etc. In [19] we provide a non-trivial application of the results obtained here to construct local models in Poisson geometry and to obtain linearization results around Poisson submanifolds.

Differentiable stratified groupoids and a de Rham theorem for inertia spaces

We introduce the notions of differentiable groupoids and differentiable stratified groupoids, generalizations of Lie groupoids in which the spaces of objects and arrows have the structures of differentiable spaces, respectively differentiable stratified spaces, compatible with the groupoid structure. After studying basic properties of these groupoids including Morita equivalence, we prove a de Rham theorem for proper locally contractible differentiable stratified groupoids. We then focus on the study of the inertia groupoid associated to a proper Lie groupoid. We show that the loop and the inertia space of a proper Lie groupoid can be endowed with a natural Whitney (b)-regular stratification, which we call the orbit Cartan type stratification. Endowed with this stratification, the inertia groupoid of a proper Lie groupoid becomes a locally contractible differentiable stratified groupoid.

On the Hochschild homology of proper Lie groupoids

We study the Hochschild homology of the convolution algebra of a proper Lie groupoid by introducing a convolution sheaf over the space of orbits. We develop a localization result for the associated Hochschild homology sheaf, and we prove that the Hochschild homology sheaf at each stalk is quasi-isomorphic to the stalk at the origin of the Hochschild homology of the convolution algebra of its linearization, which is the transformation groupoid of a linear action of a compact isotropy group on a vector space. We then explain Brylinski’s ansatz to compute the Hochschild homology of the transformation groupoid of a compact group action on a manifold. We verify Brylinski’s conjecture for the case of smooth circle actions that the Hochschild homology is given by basic relative forms on the associated inertia space.

Hodge theory for non-compact G-manifolds with boundary

We develop a generalized Hodge theory of invariant differential forms on non-compact manifolds with boundary which admit proper cocompact Lie group actions. As applications, we discuss properties of the Euler characteristic, the relative Euler characteristic and the Hopf cyclic cohomology for proper cocompact actions.

A vanishing theorem for the Mathai-Zhang index

We establish a vanishing theorem for spin manifolds admitting proper cocompact Lie group actions. To be more precise, when the connected component of the identity element in the action group is non-unimodular, we prove that the Mathai-Zhang index [Adv. Math. 225 (2010), pp. 1224–1247] vanishes.

Orbifold Euler characteristics of non‐orbifold groupoids

For a finitely presented discrete group $\Gamma$, we introduce two generalizations of the orbifold Euler characteristic and $\Gamma$-orbifold Euler characteristic to a class of proper topological groupoids large enough to include all cocompact proper Lie groupoids. The $\Gamma$-Euler characteristic is defined as an integral with respect to the Euler characteristic over the orbit space of the groupoid, and the $\Gamma$-inertia Euler characteristic is the usual Euler characteristic of the $\Gamma$-inertia space associated to the groupoid. A key ingredient is the application of o-minimal structures to study orbit spaces of topological groupoids. Our main result is that the $\Gamma$-Euler characteristic and $\Gamma$-inertia Euler characteristic coincide and generalize the higher-order orbifold Euler characteristics of Gusein-Zade, Luengo, and Melle-Hern\'{a}ndez from the case of a translation groupoid by a compact Lie group and $\Gamma = \mathbb{Z}^\ell$. By realizing the $\Gamma$-Euler characteristic as the usual Euler characteristic of a topological space, we demonstrate that it is Morita invariant in the category of topological groupoids and satisfies familiar properties of the classical Euler characteristic. We give an additional formulation of the $\Gamma$-Euler characteristic for a cocompact proper Lie groupoid in terms of a finite covering by orbispace charts. In the case that the groupoid is an abelian extension of a translation groupoid by a bundle of groups, we relate the $\Gamma$-Euler characteristics to those of the translation groupoid and bundle of groups.

Proper Lie automorphisms of incidence algebras

AbstractLet X be a finite connected poset and K a field. We study the question, when all Lie automorphisms of the incidence algebra I(X, K) are proper. Without any restriction on the length of X, we find only a sufficient condition involving certain equivalence relation on the set of maximal chains of X. For some classes of posets of length one, such as finite connected crownless posets (i.e., without weak crown subposets), crowns, and ordinal sums of two anti-chains, we give a complete answer.

On invariant linearization of Lie groupoids

The linearization theorem for proper Lie groupoids organizes and generalizes several results for classic geometries. Despite the various approaches and recent works on the subject, the problem of understanding invariant linearization remains somehow open. We address it here, by first giving a counter-example to a previous conjecture, and then proving a sufficient criterion that uses compatible complete metrics and covers the case of proper group actions. We also show a partial converse that fixes and extends previous results in the literature.

Polar actions on Damek-Ricci spaces

A proper isometric Lie group action on a Riemannian manifold is called polar if there exists a closed connected submanifold which meets all orbits orthogonally. In this article we study polar actions on Damek-Ricci spaces. We prove criteria for isometric actions on Damek-Ricci spaces to be polar, find examples and give some partial classifications of polar actions on Damek-Ricci spaces. In particular, we show that non-trivial polar actions exist on all Damek-Ricci spaces.

Euler-like vector fields, normal forms, and isotropic embeddings

As shown by Bursztyn et al. (2019), germs of tubular neighborhood embeddings for submanifolds N⊆M are in one–one correspondence with germs of Euler-like vector fields near N. In many contexts, this reduces the proof of ‘normal forms results’ for geometric structures to the construction of an Euler-like vector field compatible with the given structure. We illustrate this principle in a variety of examples, including the Morse–Bott lemma, Weinstein’s Lagrangian embedding theorem, and Zung’s linearization theorem for proper Lie groupoids. In the second part of this article, we extend the theory to a weighted context, with an application to isotropic embeddings.

Proper Lie groupoids are real analytic

Abstract We show that any proper Lie groupoid admits a compatible real analytic structure. Our proof hinges on a Weyl unitary trick of sorts for appropriate local holomorphic groupoids.

Lie algebroids, gauge theories, and compatible geometrical structures

The construction of gauge theories beyond the realm of Lie groups and algebras leads one to consider Lie groupoids and algebroids equipped with additional geometrical structures which, for gauge invariance of the construction, need to satisfy particular compatibility conditions. This paper is supposed to analyze these compatibilities from a mathematical perspective.In particular, we show that the compatibility of a connection with a Lie algebroid that one finds is the Cartan condition, introduced previously by A. Blaom. For the metric on the base [Formula: see text] of a Lie algebroid equipped with any connection, we show that the compatibility suggested from gauge theories implies that the foliation induced by the Lie algebroid becomes a Riemannian foliation. Building upon a result of del Hoyo and Fernandes, we prove, furthermore, that every Lie algebroid integrating to a proper Lie groupoid admits a compatible Riemannian base. We also consider the case where the base is equipped with a compatible symplectic or generalized Riemannian structure.

Resolutions of Proper Riemannian Lie Groupoids

Abstract In this paper we prove that every proper Lie groupoid admits a regularization to a regular proper Lie groupoid. When equipped with a Riemannian metric, we show that it admits regularization to a regular Riemannian proper Lie groupoid, arbitrarily close to the original one in the Gromov–Hausdorff distance between the quotient spaces. We construct the regularization via a successive blow-up construction on a proper Lie groupoid. We also prove that our construction of the regularization is invariant under Morita equivalence of groupoids, showing that it is a desingularization of the underlying differentiable stack.

Regular Cartan groupoids and longitudinal representations

With the intent of laying the groundwork for a program that aims at explicitly describing the space of Cartan (i.e. multiplicative) connections on a general proper Lie groupoid, we begin to investigate the space of such connections in the regular case. We point out that there is a close relationship between Cartan connections on a proper regular groupoid and representations of the groupoid on its own longitudinal bundle (i.e. on the vector distribution tangent to its orbits). This observation enables us to reduce the original problem to a simpler one. We carry out a prospective study of the latter problem, and apply the resulting analysis to produce a number of examples in rank two which serve to illustrate the diversity of the possible obstructions to the existence of multiplicative connections.

Hopf cyclic cohomology and Hodge theory for proper actions on complex manifolds

We introduce two Hopf algebroids associated to a proper and holomorphic Lie group action on a complex manifold. We prove that the cyclic cohomology of each Hopf algebroid is equal to the Dolbeault cohomology of invariant differential forms. When the action is cocompact, we develop a generalized complex Hodge theory for the Dolbeault cohomology of invariant differential forms. We prove that every cyclic cohomology class of these two Hopf algebroids can be represented by a generalized harmonic form. This implies that the space of cyclic cohomology of each Hopf algebroid is finite dimensional. As an application of the techniques developed in this paper, we generalize the Serre duality and prove a Kodaira type vanishing theorem.

Orbispaces as differentiable stratified spaces

We present some features of the smooth structure and of the canonical stratification on the orbit space of a proper Lie groupoid. One of the main features is that of Morita invariance of these structures—it allows us to talk about the canonical structure of differentiable stratified space on the orbispace (an object analogous to a separated stack in algebraic geometry) presented by the proper Lie groupoid. The canonical smooth structure on an orbispace is studied mainly via Spallek’s framework of differentiable spaces, and two alternative frameworks are then presented. For the canonical stratification on an orbispace, we extend the similar theory coming from proper Lie group actions. We make no claim to originality. The goal of these notes is simply to give a complementary exposition to those available and to clarify some subtle points where the literature can sometimes be confusing, even in the classical case of proper Lie group actions.

Reduction and Integrability of Stochastic Dynamical Systems

This paper is devoted to the study of qualitative geometrical properties of stochastic dynamical systems, namely their symmetries, reduction, and integrability. In particular, we show that an SDS that is diffusion-wise symmetric with respect to a proper Lie group action can be diffusion-wise reduced to an SDS on the quotient space. We also show necessary and sufficient conditions for an SDS to be projectable via a surjective map. We then introduce the notion of integrability of SDS’s, and extend the results on the existence and structure-preserving property of Liouville torus actions from the classical case to the case of integrable SDS’s. We also show how integrable SDS’s are related to compatible families of integrable Riemannian metrics on manifolds.

The category of reduced orbifolds in local charts

It is well-known that reduced smooth orbifolds and proper effective foliation Lie groupoids form equivalent categories. However, for certain recent lines of research, equivalence of categories is not sufficient. We propose a notion of maps between reduced smooth orbifolds and a definition of a category in terms of marked proper effective etale Lie groupoids such that the arising category of orbifolds is isomorphic (not only equivalent) to this groupoid category.