scholarly journals Complete Lie algebroid actions and the integrability of Lie algebroids

2021 ◽  
pp. 1
Author(s):  
Daniel Álvarez
Keyword(s):  
Lie Algebroids ◽  
Lie Algebroid

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 LIE ALGEBROIDS AS SPACES

2018 ◽  
Vol 19 (2) ◽  
pp. 487-535 ◽  
Author(s):  
Ryan Grady ◽  
Owen Gwilliam
Keyword(s):  
Tangent Bundle ◽  
Symplectic Structure ◽  
Vector Bundles ◽  
Natural Generalization ◽  
Lie Algebroids ◽  
Lie Algebroid ◽  
Faithful Functor ◽  
Derived Geometry

In this paper, we relate Lie algebroids to Costello’s version of derived geometry. For instance, we show that each Lie algebroid – and the natural generalization to dg Lie algebroids – provides an (essentially unique) $L_{\infty }$ space. More precisely, we construct a faithful functor from the category of Lie algebroids to the category of $L_{\infty }$ spaces. Then we show that for each Lie algebroid $L$, there is a fully faithful functor from the category of representations up to homotopy of $L$ to the category of vector bundles over the associated $L_{\infty }$ space. Indeed, this functor sends the adjoint complex of $L$ to the tangent bundle of the $L_{\infty }$ space. Finally, we show that a shifted symplectic structure on a dg Lie algebroid produces a shifted symplectic structure on the associated $L_{\infty }$ space.

scholarly journals GEOMETRICAL MECHANICS ON ALGEBROIDS

2006 ◽  
Vol 03 (03) ◽  
pp. 559-575 ◽  
Author(s):  
KATARZYNA GRABOWSKA ◽  
PAWEŁ URBAŃSKI ◽  
JANUSZ GRABOWSKI
Keyword(s):  
Vector Bundle ◽  
Lie Algebroids ◽  
Legendre Transform ◽  
Analytical Mechanics ◽  
Lie Algebroid ◽  
Noether Theorem ◽  
Geometric Framework ◽  
Hamiltonian Functions

A natural geometric framework is proposed, based on ideas of W. M. Tulczyjew, for constructions of dynamics on general algebroids. One obtains formalisms similar to the Lagrangian and the Hamiltonian ones. In contrast with recently studied concepts of Analytical Mechanics on Lie algebroids, this approach requires much less than the presence of a Lie algebroid structure on a vector bundle, but it still reproduces the main features of the Analytical Mechanics, like the Euler–Lagrange-type equations, the correspondence between the Lagrangian and Hamiltonian functions (Legendre transform) in the hyperregular cases, and a version of the Noether Theorem.

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.

Dual Structures on the Prolongations of a Lie Algebroid

2013 ◽  
Vol 59 (2) ◽  
pp. 373-390
Author(s):  
Liviu Popescu
Keyword(s):  
Covariant Derivative ◽  
Lie Algebroids ◽  
Legendre Transformation ◽  
Lie Algebroid ◽  
Symplectic Structures ◽  
Nonlinear Connection ◽  
Mechanical Structure

Abstract In the present paper we study the properties of dual structures on the prolongations of a Lie algebroid. We introduce the dynamical covariant derivative on Lie algebroids and prove that the nonlinear connection induced by a regular Lagrangian is compatible with the metric and symplectic structures. The notions of mechanical structure and semi-Hamiltonian section are introduced on the prolongation of the Lie algebroid to its dual bundle and their properties are investigated. Finally, we prove the equivalence between the metric nonlinear connection and semi-Hamiltonian section, using the Legendre transformation induced by a regular Hamiltonian.

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 Virial theorem in quasi-coordinates and Lie algebroid formalism

2014 ◽  
Vol 11 (09) ◽  
pp. 1450055 ◽  
Author(s):  
José F. Cariñena ◽  
Irina Gheorghiu ◽  
Eduardo Martínez ◽  
Patrícia Santos
Keyword(s):  
Virial Theorem ◽  
Mechanical Systems ◽  
Geometric Approach ◽  
Point Of View ◽  
Lie Algebroids ◽  
Hamiltonian Mechanics ◽  
Lie Algebroid ◽  
Lagrangian And Hamiltonian Mechanics ◽  
Generalized Virial Theorem

In this paper, the geometric approach to the virial theorem (VT) developed in [J. F. Cariñena, F. Falceto and M. F. Rañada, A geometric approach to a generalized virial theorem, J. Phys. A: Math. Theor. 45 (2012) 395210, 19 pp.] is written in terms of quasi-velocities (see [J. F. Cariñena, J. Nunes da Costa and P. Santos, Quasi-coordinates from the point of view of Lie algebroid structures, J. Phys. A: Math. Theor. 40 (2007) 10031–10048]). A generalization of the VT for mechanical systems on Lie algebroids is also given, using the geometric tools of Lagrangian and Hamiltonian mechanics on the prolongation of the Lie algebroid.

scholarly journals Higher extensions of Lie algebroids

2017 ◽  
Vol 19 (03) ◽  
pp. 1650034 ◽  
Author(s):  
Yunhe Sheng ◽  
Chenchang Zhu
Keyword(s):  
Lie Algebroids ◽  
Lie Algebroid ◽  
Courant Algebroids ◽  
Courant Algebroid ◽  
Semidirect Products

We study the extension of a Lie algebroid by a representation up to homotopy, including semidirect products of a Lie algebroid with such representations. The extension results in a higher Lie algebroid. We give exact Courant algebroids and string Lie 2-algebras as examples of such extensions. We then apply this to obtain a Lie 2-groupoid integrating an exact Courant algebroid.

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