The Atiyah-Singer index formula for subelliptic operators on contact manifolds. Part I

Erp, Erik van

doi:10.4007/annals.2010.171.1647

Cited by 46 publications

(76 citation statements)

References 17 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The Heisenberg calculus was developed by Folland-Stein [12] and Taylor [23] in the 1970s. An adaptation of the tangent groupoid to the Heisenberg calculus was developed in [8,21], which led to the index theorem for the Heisenberg calculus [9,10].…”

Section: Introductionmentioning

confidence: 99%

On the tangent groupoid of a filtered manifold

Erp

Yuncken

2017

Bull. London Math. Soc.

Self Cite

View full text Add to dashboard Cite

show abstract

Section: Introductionmentioning

confidence: 99%

On the tangent groupoid of a filtered manifold

Erp

Yuncken

2017

Bull. London Math. Soc.

Self Cite

View full text Add to dashboard Cite

show abstract

“…A deformation groupoid taking into account this inhomogeneous calculus was constructed independently in ( [98,25,27,26]) and [111,70,113]. Both these constructions are rather technical and based on higher jets.…”

Section: Inhomogeneous Pseudodifferential Calculusmentioning

confidence: 99%

Lie groupoids, pseudodifferential calculus, and index theory

Debord¹,

Skandalis²

2019

Advances in Noncommutative Geometry

View full text Add to dashboard Cite

Alain Connes introduced the use of Lie groupoids in noncommutative geometry in his pioneering work on the index theory of foliations. In the present paper, we recall the basic notion involved: groupoids, their C * -algebras, their pseudodifferential calculus... We review several recent and older advances on the involvement of Lie groupoids in noncommutative geometry. We then propose some open questions and possible developments of the subject. Lie groupoids and their operators algebrasWe refer to [78,80] for the classical definitions and constructions related to groupoids and their Lie algebroids. The construction of the C * -algebra of a groupoid is due to Jean Renault [100], one can look at his course [101] which is mainly devoted to locally compact groupoids. Lie groupoids GeneralitiesA groupoid is a small category in which every morphism is an isomorphism. Thus a groupoid G is a pair (G (0) , G (1) ) of sets together with structural morphisms:Units and arrows. The set G (0) denotes the set of objects (or units) of the groupoid, whereas the set G (1) is the set of morphisms (or arrows). The unit map u : G (0) → G (1) is the injective map which assigns to any object of G its identity morphism.The source and range maps s, r : G (1) → G (0) are (surjective) maps equal to identity in restriction toThe inverse ι : G (1) → G (1) is an involutive map which exchanges the source and range:for α ∈ G, (α −1 ) −1 = α and s(α −1 ) = r(α), where α −1 denotes ι(α)

show abstract

“…In Appendix B, I describe how the double explosion construction that I use here can be used to construct the parabolic tangent groupoid with which Erik van Erp [19] computed the indices of subelliptic operators on contact manifolds.…”

Section: 3mentioning

confidence: 99%

“…In [19], Erik van Erp studied a generalization of the Atiyah-Singer index theorem for contact manifolds using what he calls the parabolic tangent groupoid. This is inspired by Connes' tangent groupoid, but whereas Connes' tangent groupoid can be constructed by a simple explosion, the parabolic tangent groupoid requires a double explosion.…”

Section: Appendix B Relation To Van Erp's Parabolic Tangent Groupoidmentioning

confidence: 99%

Quantization of Planck’s constant

Hawkins

2016

Journal of Symplectic Geometry

View full text Add to dashboard Cite

ABSTRACT. This paper is about the role of Planck's constant,h, in the geometric quantization of Poisson manifolds using symplectic groupoids. In order to construct a strict deformation quantization of a given Poisson manifold, one can use all possible rescalings of the Poisson structure, which can be combined into a single "Heisenberg-Poisson" manifold. The new coordinate on this manifold is identified withh. I present an explicit construction for a symplectic groupoid integrating a Heisenberg-Poisson manifold and discuss its geometric quantization. I show that in cases whereh cannot take arbitrary values, this is enforced by Bohr-Sommerfeld conditions in geometric quantization.A Heisenberg-Poisson manifold is defined by linearly rescaling the Poisson structure, so I also discuss nonlinear variations and give an example of quantization of a nonintegrable Poisson manifold using a presymplectic groupoid.In appendices, I construct symplectic groupoids integrating a more general class of Heisenberg-Poisson manifolds constructed from Jacobi manifolds and discuss the parabolic tangent groupoid.

show abstract

The Atiyah-Singer index formula for subelliptic operators on contact manifolds. Part I

Cited by 46 publications

References 17 publications

On the tangent groupoid of a filtered manifold

On the tangent groupoid of a filtered manifold

Lie groupoids, pseudodifferential calculus, and index theory

Quantization of Planck’s constant

Contact Info

Product

Resources

About