Representations up to homotopy and Bottʼs spectral sequence for Lie groupoids

Abad, Camilo Arias; Crainic, Marius

doi:10.1016/j.aim.2012.12.022

Cited by 51 publications

(186 citation statements)

References 11 publications

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“…and Ω g1,g2 : E s(g2) → C t(g1) the curvature term associated to the representation (see [1,22]). The semidirect product of G with the ruth is the vector bundle V = t * C ⊕ s * E → G endowed with the VB-groupoid structure V ⇒ E given by:…”

Section: 1mentioning

confidence: 99%

“…In the next Proposition, we investigate how core sections integrate to VB-groupoids. 1 C has degree −1 and E has degree 0. 2 A vector bundle can be seen as a Lie groupoid where s = t = the bundle projection and multiplication is the fiberwise sum.…”

Section: Differentiationmentioning

confidence: 99%

“…Recall that VB-groupoids are, roughly, vector bundles in the category of Lie groupoids. They naturally extend the notion of Lie-groupoid representations by enconding the information of representations up to homotopy [22] (see also [1]), with the tangent bundle playing the role of the adjoint representation. The infinitesimal counterparts of VB-groupoids are known as VB-algebroids.…”

Section: Introductionmentioning

confidence: 99%

“…Cartan connection on G imposes strong restrictions on the Lie groupoid. Indeed, assuming G has source 1-connected fibers and M is compact, simply connected, the existence of a Cartan connection is equivalent to G being the action groupoid corresponding to a Lie group action on M (see[1, Rem. 2.14]).…”

mentioning

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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Section: 1mentioning

confidence: 99%

Section: Differentiationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

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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“…Two problems come out in this case: first, for Lie algebroids and Lie groupoids, there is not a normal adjoint representation. People solve this by using the first jet algeboids and the first jet groupoids [6,4] or by constructing a representation up to homotopy [6,1]. We shall use the jet groupoid coadjoint acting on A * to realize the symplectic leaves.…”

Section: Introductionmentioning

confidence: 99%

Coadjoint orbits of Lie groupoids

Lang¹,

Liu²

2018

Journal of Geometry and Physics

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For a Lie groupoid G with Lie algebroid A, we realize the symplectic leaves of the Lie-Poisson structure on A * as orbits of the affine coadjoint action of the Lie groupoid J G ⋉ T * M on A * , which coincide with the groupoid orbits of the symplectic groupoid T * G over A * . It is also shown that there is a fiber bundle structure on each symplectic leaf. In the case of gauge groupoids, a symplectic leaf is the universal phase space for a classical particle in a Yang-Mills field. Coadjoint representation of Lie groupoidsFor a Lie groupoid, there is no natural way to define an ordinary adjoint representation. One approach is to construct a representation up to homotopy of the Lie groupoid G on the normal complex A ⊖ T M [6], where A → M is the Lie algebroid of G. See also [1] for more detailed discussion. On the other hand, unlike the Lie groupoid G itself, the first jet groupoid J G has a natural representation on the normal complex A ⊖ T M [6,4]. We refer to [12] for the general theory of Lie groupoids and algebroids.Here we shall first recall the definitions of adjoint and coadjoint representations given by the second idea. Then we discuss the mechanical properties of the coadjoint representation.Given a Lie groupoid G ⇒ M and a surjective smooth map J : P → M , a left action of G on P along the map J, called the moment map, is a smooth map G × M P → P, (g, p) → g · p satisfying the following:where G × M P = {(g, p)|r(g) = J(p)}. Then P is called a left G-space. A representation of a groupoid G ⇒ M is a vector bundle E over M with a linear action of G on E, i.e. for each arrow g : x → y, the induced map g· : E x → E y on the fibers is a linear isomorphism.Let G be a Lie groupoid with Lie algebroid A. A bisection of G is a splitting b : M → G of the source map r with the property that φ b := l • b : M → M is a diffeomorphism. The bisections of G form a group Bis(G) with the multiplication and inverse given by

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On Deformations of Lie Algebroids

Sheng

2011

Results. Math.

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For any Lie algebroid A, its 1-jet bundle JA is a Lie algebroid naturally and there is a representation π : JA −→ DA. Denote by d J the corresponding coboundary operator. In this paper, we realize the deformation cohomology of a Lie algebroid A introduced by M. Crainic and I. Moerdijk as the cohomology of a subcomplex (Γ(Hom(∧ • JA, A) DA ), d J ) of the cochain complex (Γ(Hom(∧ • JA, A)), d J ). (2000). Primary 17B65, Secondary 18B40, 58H05. Mathematics Subject Classification

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Representations up to homotopy and Bottʼs spectral sequence for Lie groupoids

Cited by 51 publications

References 11 publications

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

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

Coadjoint orbits of Lie groupoids

On Deformations of Lie Algebroids

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