LIE GROUPOIDS AND LIE ALGEBROIDS IN DIFFERENTIAL GEOMETRY

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.

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Lie groupoids and Lie algebroids

Lie Groupoids and Lie Algebroids in Differential Geometry ◽

10.1017/cbo9780511661839.004 ◽

1987 ◽

pp. 82-184 ◽

Cited By ~ 1

Keyword(s):

Lie Algebroids ◽

Lie Groupoids

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Transitive double Lie algebroids via core diagrams

The Journal of Geometric Mechanics ◽

10.3934/jgm.2021023 ◽

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

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 transitive. 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.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.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.

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Information Geometry on Groupoids: The Case of Singular Metrics

Open Systems & Information Dynamics ◽

10.1142/s1230161220500158 ◽

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.

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Principal Actions of Stacky Lie Groupoids

International Mathematics Research Notices ◽

10.1093/imrn/rny142 ◽

2018 ◽

Vol 2020 (16) ◽

pp. 5055-5125

Author(s):

Henrique Bursztyn ◽

Francesco Noseda ◽

Chenchang Zhu

Keyword(s):

Principal Bundle ◽

Morita Equivalence ◽

Lie Algebroids ◽

Lie Groupoids ◽

Differentiable Stack ◽

Principal Actions

Abstract Stacky Lie groupoids are generalizations of Lie groupoids in which the “space of arrows” of the groupoid is a differentiable stack. In this paper, we consider actions of stacky Lie groupoids on differentiable stacks and their associated quotients. We provide a characterization of principal actions of stacky Lie groupoids, that is, actions whose quotients are again differentiable stacks in such a way that the projection onto the quotient is a principal bundle. As an application, we extend the notion of Morita equivalence of Lie groupoids to the realm of stacky Lie groupoids, providing examples that naturally arise from non-integrable Lie algebroids.

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