scholarly journals Riemannian metrics on Lie groupoids

2018 ◽  
Vol 2018 (735) ◽  
pp. 143-173 ◽  
Author(s):  
Matias del Hoyo ◽  
Rui Loja Fernandes
Keyword(s):  
Riemannian Metrics ◽  
Exponential Map ◽  
Important Class ◽  
Lie Groupoids ◽  
Lie Groupoid ◽  
Linearization Theorem

AbstractWe introduce a notion of metric on a Lie groupoid, compatible with multiplication, and we study its properties. We show that many families of Lie groupoids admit such metrics, including the important class of proper Lie groupoids. The exponential map of these metrics allows us to establish a linearization theorem for Riemannian groupoids, obtaining both a simpler proof and a stronger version of the Weinstein–Zung linearization theorem for proper Lie groupoids. This new notion of metric has a simplicial nature which will be explored in future papers of this series.

scholarly journals On local integration of Lie brackets

2020 ◽  
Vol 2020 (760) ◽  
pp. 267-293 ◽  
Author(s):  
Alejandro Cabrera ◽  
Ioan Mărcuţ ◽  
María Amelia Salazar
Keyword(s):  
Vector Field ◽  
Lie Algebroids ◽  
Lie Algebroid ◽  
Lie Theory ◽  
Lie Groupoids ◽  
Lie Groupoid ◽  
Finite Dimensional ◽  
Local Integration ◽  
Lie Brackets ◽  
Complete Account

AbstractWe 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.

scholarly journals Deformations of Lie Groupoids

2018 ◽  
Vol 2020 (21) ◽  
pp. 7662-7746 ◽  
Author(s):  
Marius Crainic ◽  
João Nuno Mestre ◽  
Ivan Struchiner
Keyword(s):  
Normal Form ◽  
Adjoint Representation ◽  
Lie Groupoids ◽  
Lie Groupoid ◽  
Fundamental Properties

Abstract We study deformations of Lie groupoids by means of the cohomology which controls them. This cohomology turns out to provide an intrinsic model for the cohomology of a Lie groupoid with values in its adjoint representation. We prove several fundamental properties of the deformation cohomology including Morita invariance, a van Est theorem, and a vanishing result in the proper case. Combined with Moser’s deformation arguments for groupoids, we obtain several rigidity and normal form results.

scholarly journals Lie groupoids and algebroids applied to the study of uniformity and homogeneity of Cosserat media

2018 ◽  
Vol 15 (08) ◽  
pp. 1830003 ◽  
Author(s):  
Víctor Manuel Jiménez ◽  
Manuel de León ◽  
Marcelo Epstein
Keyword(s):  
Second Order ◽  
Lie Algebroid ◽  
Lie Groupoids ◽  
Lie Groupoid ◽  
Text Structures ◽  
Cosserat Medium ◽  
Homogeneity Property ◽  
Definition Of ◽  
Cosserat Media ◽  
Natural Way

A Lie groupoid, called second-order non-holonomic material Lie groupoid, is associated in a natural way to any Cosserat medium. This groupoid is used to give a new definition of homogeneity which does not depend on a material archetype. The corresponding Lie algebroid, called second-order non-holonomic material Lie algebroid, is used to characterize the homogeneity property of the material. We also relate these results with the previously obtained ones in terms of non-holonomic second-order [Formula: see text]-structures.

scholarly journals Transitive double Lie algebroids via core diagrams

2021 ◽  
Vol 13 (3) ◽  
pp. 403
Author(s):  
Madeleine Jotz Lean ◽  
Kirill C. H. Mackenzie
Keyword(s):  
Lie Algebroids ◽  
Lie Algebroid ◽  
Lie Groupoids ◽  
Base Manifold ◽  
Lie Groupoid ◽  
The Core ◽  
Fixed Base ◽  
The One

<p style='text-indent:20px;'>The core diagram of a double Lie algebroid consists of the core of the double Lie algebroid, together with the two core-anchor maps to the sides of the double Lie algebroid. If these two core-anchors are surjective, then the double Lie algebroid and its core diagram are called <i>transitive</i>. This paper establishes an equivalence between transitive double Lie algebroids, and transitive core diagrams over a fixed base manifold. In other words, it proves that a transitive double Lie algebroid is completely determined by its core diagram.</p><p style='text-indent:20px;'>The comma double Lie algebroid associated to a morphism of Lie algebroids is defined. If the latter morphism is one of the core-anchors of a transitive core diagram, then the comma double algebroid can be quotiented out by the second core-anchor, yielding a transitive double Lie algebroid, which is the one that is equivalent to the transitive core diagram.</p><p style='text-indent:20px;'>Brown's and Mackenzie's equivalence of transitive core diagrams (of Lie groupoids) with transitive double Lie groupoids is then used in order to show that a transitive double Lie algebroid with integrable sides and core is automatically integrable to a transitive double Lie groupoid.</p>

scholarly journals Information Geometry on Groupoids: The Case of Singular Metrics

2020 ◽  
Vol 27 (03) ◽  
pp. 2050015
Author(s):  
Katarzyna Grabowska ◽  
Janusz Grabowski ◽  
Marek Kuś ◽  
Giuseppe Marmo
Keyword(s):  
General Setting ◽  
Information Geometry ◽  
Potential Functions ◽  
Lie Algebroids ◽  
Lie Algebroid ◽  
Lie Groupoids ◽  
Riemannian Foliations ◽  
Lie Groupoid ◽  
Singular Metrics

We use the general setting for contrast (potential) functions in statistical and information geometry provided by Lie groupoids and Lie algebroids. The contrast functions are defined on Lie groupoids and give rise to two-forms and three-forms on the corresponding Lie algebroid. We study the case when the two-form is degenerate and show how in sufficiently regular cases one reduces it to a pseudometric structures. Transversal Levi-Civita connections for Riemannian foliations are generalized to the Lie groupoid/Lie algebroid case.

scholarly journals Material groupoids and algebroids

2018 ◽  
Vol 24 (3) ◽  
pp. 796-806 ◽  
Author(s):  
Marcelo Epstein ◽  
Manuel de León
Keyword(s):  
Differential Geometry ◽  
Continuum Mechanics ◽  
Index Of Refraction ◽  
Lie Algebroid ◽  
Lie Groupoids ◽  
Material Body ◽  
Material Defects ◽  
Lie Groupoid ◽  
Constitutive Properties

Lie groupoids and their associated algebroids arise naturally in the study of the constitutive properties of continuous media. Thus, continuum mechanics and differential geometry illuminate each other in a mutual entanglement of theory and applications. Given any material property, such as the elastic energy or an index of refraction, affected by the state of deformation of the material body, one can automatically associate to it a groupoid. Under conditions of differentiability, this material groupoid is a Lie groupoid. Its associated Lie algebroid plays an important role in the determination of the existence of material defects, such as dislocations. This paper presents a rather intuitive treatment of these ideas.

scholarly journals Equivariant K-theory, groupoids and proper actions

2011 ◽  
Vol 9 (3) ◽  
pp. 475-501
Author(s):  
Jose Cantarero
Keyword(s):  
Vector Bundles ◽  
Cohomology Theory ◽  
Dimensional Vector ◽  
Lie Groupoids ◽  
Proper Actions ◽  
Lie Groupoid ◽  
Finite Dimensional ◽  
Cw Complexes ◽  
K Theory ◽  
Completion Theorem

AbstractIn this paper we define complex equivariant K-theory for actions of Lie groupoids using finite-dimensional vector bundles. For a Bredon-compatible Lie groupoid , this defines a periodic cohomology theory on the category of finite -CW-complexes. We also establish an analogue of the completion theorem of Atiyah and Segal. Some examples are discussed.

scholarly journals Universal connections on Lie groupoids

2003 ◽  
Vol 2003 (23) ◽  
pp. 1465-1480
Author(s):  
Efstathios Vassiliou ◽  
Apostolos Nikolopoulos
Keyword(s):  
Vector Bundles ◽  
General Construction ◽  
Lie Groupoids ◽  
Principal Bundles ◽  
Lie Groupoid ◽  
Linear Connections ◽  
Connections On Principal Bundles

Given a Lie groupoidΩ, we construct a groupoidJ1Ωequipped with a universal connection from which all the connections ofΩare obtained by certain pullbacks. We show that this general construction leads to universal connections on principal bundles (considered by García (1972)) and universal linear connections on vector bundles (ultimately related with those of Cordero et al. (1989)).

scholarly journals On Rham cohomology of locally trivial Lie groupoids over triangulated manifolds

2020 ◽  
Vol 13 (4) ◽  
pp. 116-125
Author(s):  
Jose R. Oliveira
Keyword(s):  
Simplicial Complex ◽  
Smooth Manifold ◽  
Piecewise Smooth ◽  
Ambient Space ◽  
Lie Algebroid ◽  
Lie Groupoids ◽  
De Rham Cohomology ◽  
Lie Groupoid ◽  
De Rham ◽  
Finite Simplicial Complex

Based on the isomorphism between Lie algebroid cohomology and piecewise smooth cohomology of a transitive Lie algebroid, it is proved that the Rham cohomology of a locally trivial Lie groupoid G on a smooth manifold M is isomorphic to the piecewise Rham cohomology of G, in which G and M are manifolds without boundary and M is smoothly triangulated by a finite simplicial complex K such that, for each simplex ∆ of K, the inverse images of ∆ by the source and target mappings of G are transverses submanifolds in the ambient space G. As a consequence, it is shown that the piecewise de Rham cohomology of G does not depend on the triangulation of the base.

Local index theory for operators associated with Lie groupoid actions

2020 ◽  
pp. 1-45
Author(s):  
Denis Perrot
Keyword(s):  
Fixed Point ◽  
Index Theory ◽  
Cyclic Cohomology ◽  
Lie Groupoids ◽  
Fixed Point Set ◽  
Local Index ◽  
Lie Groupoid ◽  
Residue Formula ◽  
Isometric Actions ◽  
Point Set

We develop a local index theory for a class of operators associated with non-proper and non-isometric actions of Lie groupoids on smooth submersions. Such actions imply the existence of a short exact sequence of algebras, relating these operators to their non-commutative symbol. We then compute the connecting map induced by this extension on periodic cyclic cohomology. When cyclic cohomology is localized at appropriate isotropic submanifolds of the groupoid in question, we find that the connecting map is expressed in terms of an explicit Wodzicki-type residue formula, which involves the jets of non-commutative symbols at the fixed-point set of the action.

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