Modular Classes of Lie Algebroid Morphisms

Kosmann–Schwarzbach, Yvette; Laurent-Gengoux, Camille; Weinstein, A. J.

doi:10.1007/s00031-008-9032-y

Cited by 23 publications

(38 citation statements)

References 24 publications

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“…The extension of the definition and properties of the modular class to the case of Lie algebroid morphisms that are not necessarily base-preserving is the subject of the article [73]. Let us briefly review this general case.…”

Section: General Morphisms Of Lie Algebroidsmentioning

confidence: 99%

“…Another theorem in [73] states that the pull-back of a Lie algebroid by a transverse map, in particular by a submersion, gives rise to a morphism with vanishing modular class.…”

Section: Remark 4 It Was Proved By Chen and Liumentioning

confidence: 99%

“…Formula (6.1) can also be obtained from the more general computation of the modular class of a Lie algebroid morphism with constant rank and unimodular kernel in [73].…”

Section: The Regular Casementioning

confidence: 99%

See 2 more Smart Citations

Poisson Manifolds, Lie Algebroids, Modular Classes: a Survey

Kosmann–Schwarzbach¹

2008

SIGMA

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Section: General Morphisms Of Lie Algebroidsmentioning

confidence: 99%

“…Another theorem in [73] states that the pull-back of a Lie algebroid by a transverse map, in particular by a submersion, gives rise to a morphism with vanishing modular class.…”

Section: Remark 4 It Was Proved By Chen and Liumentioning

confidence: 99%

See 1 more Smart Citation

Poisson Manifolds, Lie Algebroids, Modular Classes: a Survey

Kosmann–Schwarzbach¹

2008

SIGMA

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“…To this end we first have to recall some results from [16] on the modular class, see also [44,32,31] as well as [26] for a more algebraic approach. However, we shall use a slightly different presentation.…”

Section: The Trace In the Unimodular Casementioning

confidence: 99%

Deformation Quantization of Poisson Structures Associated to Lie Algebroids

Neumaier

Waldmann

2009

SIGMA

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Abstract. In the present paper we explicitly construct deformation quantizations of certain Poisson structures on E * , where E −→ M is a Lie algebroid. Although the considered Poisson structures in general are far from being regular or even symplectic, our construction gets along without Kontsevich's formality theorem but is based on a generalized Fedosov construction. As the whole construction merely uses geometric structures of E we also succeed in determining the dependence of the resulting star products on these data in finding appropriate equivalence transformations between them. Finally, the concreteness of the construction allows to obtain explicit formulas even for a wide class of derivations and selfequivalences of the products. Moreover, we can show that some of our products are in direct relation to the universal enveloping algebra associated to the Lie algebroid. Finally, we show that for a certain class of star products on E * the integration with respect to a density with vanishing modular vector field defines a trace functional.

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“…Here, an extension to groupoids of the relative modular class for Lie algebroid morphisms in [12] should come into play.…”

Section: Introductionmentioning

confidence: 99%

The Volume of a Differentiable Stack

Weinstein

2009

Lett Math Phys

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Abstract. We extend the notion of the cardinality of a discrete groupoid (equal to the Euler characteristic of the corresponding discrete orbifold) to the setting of Lie groupoids. Since this quantity is an invariant under equivalence of groupoids, we call it the volume of the associated stack rather than of the groupoid itself. Since there is no natural measure in the smooth case like the counting measure in the discrete case, we need extra data to define the volume. This data has the form of an invariant section of a natural line bundle over the base of the groupoid. Invariant sections of a square root of this line bundle constitute an "intrinsic Hilbert space" of the stack. Mathematics Subject Classification (2000). Primary 58H05; Secondary 53D17.

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Modular Classes of Lie Algebroid Morphisms

Cited by 23 publications

References 24 publications

Poisson Manifolds, Lie Algebroids, Modular Classes: a Survey

Poisson Manifolds, Lie Algebroids, Modular Classes: a Survey

Deformation Quantization of Poisson Structures Associated to Lie Algebroids

The Volume of a Differentiable Stack

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