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 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 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 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.

scholarly journals Fiberwise linear differential operators

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Fabrizio Pugliese ◽  
Giovanni Sparano ◽  
Luca Vitagliano
Keyword(s):  
Vector Bundle ◽  
Differential Operator ◽  
Line Bundle ◽  
Differential Operators ◽  
Linear Differential Operator ◽  
Lie Algebroid ◽  
Lie Groupoid ◽  
Linear Differential Operators ◽  
Linear Differential ◽  
Definition Of

Abstract We define a new notion of fiberwise linear differential operator on the total space of a vector bundle E. Our main result is that fiberwise linear differential operators on E are equivalent to (polynomial) derivations of an appropriate line bundle over E ∗ {E^{\ast}} . We believe this might represent a first step towards a definition of multiplicative (resp. infinitesimally multiplicative) differential operators on a Lie groupoid (resp. a Lie algebroid). We also discuss the linearization of a differential operator around a submanifold.

scholarly journals A note on symmetries of diffusions within a martingale problem approach

2019 ◽  
Vol 19 (02) ◽  
pp. 1950011 ◽  
Author(s):  
Francesco C. De Vecchi ◽  
Paola Morando ◽  
Stefania Ugolini
Keyword(s):  
Diffusion Processes ◽  
Martingale Problem ◽  
Second Order ◽  
Lie Symmetries ◽  
Definition Of

A geometric reformulation of the martingale problem associated with a set of diffusion processes is proposed. This formulation, based on second-order geometry and Itô integration on manifolds, allows us to give a natural and effective definition of Lie symmetries for diffusion processes.

A Large Scale Cross-Validation of Second-Order Personality Structure Defined by the 16PF

1986 ◽  
Vol 59 (2) ◽  
pp. 683-693 ◽  
Author(s):  
Samuel E. Krug ◽  
Edgar F. Johns
Keyword(s):  
Large Scale ◽  
Common Factor ◽  
Geographic Location ◽  
Second Order ◽  
Personality Factor ◽  
Personality Factor Questionnaire ◽  
Normal Males ◽  
Definition Of ◽  
High Degree ◽  
Factor Estimation

The second-order factors structure of the 16 Personality Factor Questionnaire (16PF) was cross-validated on a large sample ( N = 17,381) of normal males and females. Subjects were sampled across a broad range of ages, socioeconomic levels, education, geographic location, and ethnicity. The purposes of this investigation were (1) to provide a precise definition of 16PF second-order factor structure, (2) to shed additional light on the nature of two second-order factors that have been previously identified but described as “unstable” and “poorly reproduced,” and (3) to determine the extent to which common factor estimation formulas for men and women would prove satisfactory for applied work. The resulting solutions were congruent with previous studies and showed a high degree of simple structure. Support was provided for one, but not both, of the two additional second-order factors. Results also supported the use of simplified estimation formulas for applied use.

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