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 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 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 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 GEOMETRIC QUANTIZATION OF HAMILTONIAN ACTIONS OF LIE ALGEBROIDS AND LIE GROUPOIDS

2007 ◽  
Vol 04 (03) ◽  
pp. 389-436 ◽  
Author(s):  
ROGIER BOS
Keyword(s):  
Geometric Quantization ◽  
Symplectic Manifolds ◽  
Lie Algebroids ◽  
Unitary Representations ◽  
Reduction Theorem ◽  
Orbit Method ◽  
Lie Algebroid ◽  
Lie Groupoids ◽  
Quantization Procedure ◽  
Hamiltonian Actions

We construct Hermitian representations of Lie algebroids and associated unitary representations of Lie groupoids by a geometric quantization procedure. For this purpose, we introduce a new notion of Hamiltonian Lie algebroid actions. The first step of our procedure consists of the construction of a prequantization line bundle. Next, we discuss a version of Kähler quantization suitable for this setting. We proceed by defining a Marsden–Weinstein quotient for our setting and prove a "quantization commutes with reduction" theorem. We explain how our geometric quantization procedure relates to a possible orbit method for Lie groupoids. Our theory encompasses the geometric quantization of symplectic manifolds, Hamiltonian Lie algebra actions, actions of bundles of Lie groups, and foliations, as well as some general constructions from differential geometry.

scholarly journals Integrable lifts for transitive Lie algebroids

2018 ◽  
Vol 29 (09) ◽  
pp. 1850062 ◽  
Author(s):  
Iakovos Androulidakis ◽  
Paolo Antonini
Keyword(s):  
Extra Dimensions ◽  
Lie Algebroids ◽  
The Other ◽  
Lie Algebroid ◽  
Base Manifold ◽  
Lie Groupoid ◽  
Finite Dimensional ◽  
Other Hand ◽  
De Rham ◽  
Simply Connected

Inspired by the work of Molino, we show that the integrability obstruction for transitive Lie algebroids can be made to vanish by adding extra dimensions. In particular, we prove that the Weinstein groupoid of a non-integrable transitive and abelian Lie algebroid is the quotient of a finite-dimensional Lie groupoid. Two constructions as such are given: First, explaining the counterexample to integrability given by Almeida and Molino, we see that it can be generalized to the construction of an “Almeida–Molino” integrable lift when the base manifold is simply connected. On the other hand, we notice that the classical de Rham isomorphism provides a universal integrable algebroid. Using it we construct a “de Rham” integrable lift for any given transitive Abelian Lie algebroid.

scholarly journals ON SYMPLECTIC DOUBLE GROUPOIDS AND THE DUALITY OF POISSON GROUPOIDS

1999 ◽  
Vol 10 (04) ◽  
pp. 435-456 ◽  
Author(s):  
K. C. H. MACKENZIE
Keyword(s):  
Lie Algebroids ◽  
Lie Algebroid ◽  
Lie Groupoid ◽  
The Core ◽  
Double Groupoid ◽  
Double Base ◽  
Double Groupoids

We prove that the cotangent of a double Lie groupoid S has itself a double groupoid structure with sides the duals of associated Lie algebroids, and double base the dual of the Lie algebroid of the core of S. Using this, we prove a result outlined by Weinstein in 1988, that the side groupoids of a general symplectic double groupoid are Poisson groupoids in duality. Further, we prove that any double Lie groupoid gives rise to a pair of Poisson groupoids (and thus of Lie bialgebroids) in duality. To handle the structures involved effectively we extend to this context the dualities and canonical isomorphisms for tangent and cotangent structures of the author and Ping Xu.

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 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.

scholarly journals Lie algebroids, gauge theories, and compatible geometrical structures

2019 ◽  
Vol 31 (04) ◽  
pp. 1950015 ◽  
Author(s):  
Alexei Kotov ◽  
Thomas Strobl
Keyword(s):  
Gauge Invariance ◽  
Gauge Theories ◽  
Riemannian Foliation ◽  
Riemannian Structure ◽  
Lie Algebroid ◽  
Lie Groupoids ◽  
Compatibility Conditions ◽  
Geometrical Structures ◽  
Lie Groupoid ◽  
Lie Groups And Algebras

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.

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