Vector bundles over Lie groupoids and algebroids

Abstract. The main goal of this work is to introduce a natural notion of ideal in a Lie algebroid, the "infinitesimal ideal systems". Ideals in Lie algebras and the Bott connection associated to involutive subbundles of tangent bundles are special cases. The definition of these objects is motivated by the infinitesimal description of multiplicative distributions on Lie groupoids, which are just ideals in the Lie group case. Several examples of infinitesimal ideal systems are presented, and the quotient of a Lie algebroid by an infinitesimal ideal system is shown to be a Lie algebroid.

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“…This is done in [3] using the characterization of vector bundles via homogeneous structures (see [11]). …”

Section: Furthermore If Ementioning

confidence: 99%

Foliated groupoids and infinitesimal ideal systems

Lean

Ortiz

2014

Indagationes Mathematicae

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“…These relations can be used as axioms to define a VB-algebroid structure on a double vector bundle as was done in [23]. From this point of view, it follows directly from the definition that formulas (2.8) extend to arbitrary VBalgebroids giving a representation on ∂ : C → E. It is important to point out that the equivalence between the definition of VB-algebroids we use here and the one given in [23] was proved in [7].…”

Section: 21mentioning

confidence: 90%

“…We shall refer to such sections as core sections. The Lie functor applied to a VB-groupoid gives rise to a VB-algebroid [7]. In the next Proposition, we investigate how core sections integrate to VB-groupoids.…”

Section: Differentiationmentioning

confidence: 99%

“…5 Proposition 5.6. ϑ ∈ Ω k (G, V) is multiplicative if and only if F ϑ is a cocycle on the differentiable cohomology of G. 5 Note that we are using the fact that the projection maps (T G ⇒ T M ) → (G ⇒ M ) and (V * ⇒ C * ) → (G ⇒ M ) fulfill the conditions for the fiber product to have a Lie groupoid structure. We refer to the appendix of [7] for a detailed discussion regarding fiber products of Lie groupoids.…”

Section: From Global To Infinitesimal Given a Vb-groupoidmentioning

confidence: 99%

“…Indeed, given a multiplicative distribution H ⊂ T G, its VB-algebroid h ⊂ T A correspond to an IM distribution via (6.4) and Proposition 6.12. Reciprocally, given an IM distribution, the corresponding VB-subalgebroid h ⊂ T A (6.5) can be integrated to a VB-subgroupoid H ⊂ T G using [7,Cor. 4.3.7].…”

Section: Im Distributionsmentioning

confidence: 99%

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Differential forms with values in VB-groupoids and its Morita invariance

Drummond

Egea

2019

Journal of Geometry and Physics

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We introduce multiplicative differential forms on Lie groupoids with values in VB-groupoids. Our main result gives a complete description of these objects in terms of infinitesimal data. By considering split VB-groupoids, we are able to present a Lie theory for differential forms on Lie groupoids with values in 2-term representations up to homotopy. We also define a differential complex whose 1-cocycles are exactly the multiplicative forms with values in VB-groupoids and study the Morita invariance of its cohomology.

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Dirac Actions and Lu’s Lie Algebroid

Meinrenken

2017

Transformation Groups

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Abstract. Poisson actions of Poisson Lie groups have an interesting and rich geometric structure. We will generalize some of this structure to Dirac actions of Dirac Lie groups. Among other things, we extend a result of Jiang-Hua-Lu, which states that the cotangent Lie algebroid and the action algebroid for a Poisson action form a matched pair. We also give a full classification of Dirac actions for which the base manifold is a homogeneous space H/K, obtaining a generalization of Drinfeld's classification for the Poisson Lie group case. Contents

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Vector bundles over Lie groupoids and algebroids

Cited by 60 publications

References 27 publications

Foliated groupoids and infinitesimal ideal systems

Foliated groupoids and infinitesimal ideal systems

Differential forms with values in VB-groupoids and its Morita invariance

Dirac Actions and Lu’s Lie Algebroid

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