Compactly supported analytic indices for Lie groupoids

2009 ◽  
Vol 4 (2) ◽  
pp. 223-262 ◽  
Author(s):  
Paulo Carrillo Rouse
Keyword(s):  
Index Theorem ◽  
Lie Groupoids ◽  
Smooth Functions ◽  
Lie Groupoid ◽  
Convolution Algebra ◽  
Compactly Supported ◽  
Potential Applications ◽  
Analytic Index ◽  
Image Position ◽  
Symbol Class

AbstractFor any Lie groupoid we construct an analytic index morphism taking values in a modified K-theory group which involves the convolution algebra of compactly supported smooth functions over the groupoid. The construction is performed by using the deformation algebra of smooth functions over the tangent groupoid constructed in [CR06]. This allows us in particular to prove a more primitive version of the Connes-Skandalis longitudinal index theorem for foliations, that is, an index theorem taking values in a group which pairs with cyclic cocycles. As another application, for D a -PDO elliptic operator with associated index ind we prove that the pairingwith τ a bounded continuous cyclic cocycle, only depends on the principal symbol class [σ(D)]∈K0. The result is completely general for étale groupoids. We discuss some potential applications to the Novikov conjecture.

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.

Role of As in the Anisotropic Positioning of Self-Assembled InAs Quantum Dots

2013 ◽  
Vol 1551 ◽  
pp. 3-9
Author(s):  
Fabrizio Arciprete ◽  
Ernesto Placidi ◽  
Rita Magri ◽  
Massimo Fanfoni ◽  
Adalberto Balzarotti ◽  
...  
Keyword(s):  
Quantum Dots ◽  
Molecular Beam ◽  
High Growth ◽  
Inas Quantum Dots ◽  
Potential Applications ◽  
Gaas Buffer Layer ◽  
Image Position ◽  
Pulsed Deposition ◽  
Deposition Mode

ABSTRACTProgress in tailoring the size, shape and positioning of Quantum Dots on the substrate is crucial for their potential applications in new optoelectronic devices for nano-photonics as well as in quantum information and computation. Using Molecular Beam Epitaxy in pulsed deposition mode we demonstrate that the nucleation of InAs Quantum Dots can be selectively guided on the GaAs(001) surface by a suitable choice of the kinetic parameters for the growth of both the GaAs buffer layer and the InAs Quantum Dots. By developing a two-species rate-equation kinetic model we show that the positioning of the Quantum Dots on only one side of mounds of the GaAs buffer can be traced back to the very small As flux gradient between the two mound slopes $\left( {\Delta F_A /F_A \approx 1 - 5\% } \right)$ caused by the proper tilting of the incoming As flux. Such gradient originates, at the relatively high growth-temperature, a net cation flow from one slope of the mound to the other that is responsible for the selective growth.

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.

Self-similar solutions of the second kind for the modified porous medium equation

1994 ◽  
Vol 5 (3) ◽  
pp. 391-403 ◽  
Author(s):  
Josephus Hulshof ◽  
Juan Luis Vazquez
Keyword(s):  
Porous Medium ◽  
Porous Medium Equation ◽  
Self Similarity ◽  
Compactly Supported ◽  
Medium Equation ◽  
Self Similar ◽  
Image Position ◽  
Similar Solutions

We construct compactly supported self-similar solutions of the modified porous medium equation (MPME)They have the formwhere the similarity exponents α and β depend on ε, m and the dimension N. This corresponds to what is known in the literature as anomalous exponents or self-similarity of the second kind, a not completely understood phenomenon. This paper performs a detailed study of the properties of the anomalous exponents of the MPME.

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 On the weakly-* dense subsets in $L^{\infty}(\Omega)$

2021 ◽  
Vol 16 ◽  
pp. 149
Author(s):  
P.I. Kogut ◽  
T.N. Rudyanova
Keyword(s):  
Weak Convergence ◽  
Density Property ◽  
Smooth Functions ◽  
Compactly Supported

In this paper we study the density property of the compactly supported smooth functions in the space $L^{\infty}(\Omega)$. We show that this set is dense with respect to the weak-* convergence in variable spaces.

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