We address one of the open problems in quantization theory recently listed by Rieffel. By developping in detail Connes' tangent groupoid principle and using previous work by Landsman, we show how to construct a strict, flabby quantization, which is moreover an asymptotic morphism and satisfies the reality and traciality constraints, on any oriented Riemannian manifold. That construction generalizes the standard Moyal rule. The paper can be considered as an introduction to quantization theory from Connes' point of view.
The purpose of this article is to study Ezra Getzler's approach to the Atiyah-Singer index theorem from the perspective of Alain Connes' tangent groupoid. We shall construct a "rescaled" spinor bundle on the tangent groupoid, define a convolution operation on its smooth, compactly supported sections, and explain how the algebra so-obtained incorporates Getzler's symbol calculus.
We construct an algebra of smooth functions over the tangent groupoid associated to any Lie groupoid. This algebra is a field of algebras over the closed interval [0, 1] which fiber at zero is the algebra of Schwartz functions over the Lie algebroid, whereas any fiber out of zero is the convolution algebra of the initial groupoid. Our motivation comes from index theory for Lie groupoids. In fact, our construction gives an intermediate algebra between the enveloping C-algebra and the convolution algebra of compactly supported functions of the tangent groupoid; and it will allows us, in a furthe…
As a step toward proving an index theorem for hypoelliptic operators Heisenberg manifolds, including those on CR and contact manifolds, we construct an analogue for Heisenberg manifolds of Connes' tangent groupoid of a manifold $M$. As it is well known for a Heisenberg manifold $(M,H)$ the relevant notion of tangent is rather that of Lie group bundle of graded 2-step nilpotent Lie groups $GM$. We then construct the tangent groupoid of $(M,H)$ as a differentiable groupoid $\cG_{H} M$ encoding the smooth deformation of $M\times M$ to $GM$. In this construction a crucial use is made of a refined …
This paper studies the infinitesimal structure of Carnot manifolds. By a Carnot manifold we mean a manifold together with a subbundle filtration of its tangent bundle which is compatible with the Lie bracket of vector fields. We introduce a notion of differential, called Carnot differential, for Carnot manifolds maps (i.e., maps that are compatible with the Carnot manifold structure). This differential is obtained as a group map between the corresponding tangent groups. We prove that, at every point, a Carnot manifold map is osculated in a very precise way by its Carnot differential at the poi…
In 1974, Folland and Stein constructed an inhomogeneous pseudo-differential calculus based on analysis on the Heisenberg group. This Heisenberg calculus was generalized by several authors, to any subbundle of the tangent bundle. van Erp and Yuncken, following Debord and Skandalis showed that this calculus can be recovered using a deformation groupoid alla tangent groupoid of Connes. Using functoriality of the deformation to the normal cone construction, we give an elementary construction of this groupoid. We then extend it to the general case of a filtration of the tangent bundle by an iterate…
A VB-groupoid is a Lie groupoid equipped with a compatible linear structure. In this paper, we describe a correspondence, up to isomorphism, between VB-groupoids and 2-term representations up to homotopy of Lie groupoids. Under this correspondence, the tangent bundle of a Lie groupoid G corresponds to the "adjoint representation" of G. The value of this point of view is that the tangent bundle is canonical, whereas the adjoint representation is not. We define a cochain complex that is canonically associated to any VB-groupoid. The cohomology of this complex is isomorphic to the groupoid cohomo…
We give an algebraic/geometric characterization of the classical pseudodifferential operators on a smooth manifold in terms of the tangent groupoid and its natural $\mathbb{R}^\times_+$-action. Specifically, we show that a properly supported semiregular distribution on $M\times M$ is the Schwartz kernel of a classical pseudodifferential operator if and only if it extends to a smooth family of distributions on the range fibres of the tangent groupoid which is homogeneous for the $\mathbb{R}^\times_+$-action modulo smooth functions. Moreover, we show that the basic properties of pseudodifferenti…
Georg-August-Universität Göttingen（哥廷根大学）；Institut Camille Jordan, Université Claude Bernard Lyon 1（里昂第一大学卡米尔·让研究所）；Fraunhofer-Institut für Energiewirtschaft und Energiesystemtechnik IEE（弗劳恩霍夫能源系统及能源经济研究所）
We define and study multiplicative connections in the tangent bundle of a Lie groupoid. Multiplicative connections are linear connections satisfying an appropriate compatibility with the groupoid structure. Our definition is natural in the sense that a linear connection on a Lie groupoid is multiplicative if and only if its torsion is a multiplicative tensor in the sense of Bursztyn-Drummond [5] and its geodesic spray is a multiplicative vector field. We identify the obstruction to the existence of a multiplicative connection. We also discuss the infinitesimal version of multiplicative connect…
For any Lie groupoid $G$, the vector bundle $g^*$ dual to the associated Lie algebroid $g$ is canonically a Poisson manifold. The (reduced) C*-algebra of $G$ (as defined by A. Connes) is shown to be a strict quantization (in the sense of M. Rieffel) of $g^*$. This is proved using a generalization of Weyl's quantization prescription on flat space. Many other known strict quantizations are a special case of this procedure; on a Riemannian manifold, one recovers Connes' tangent groupoid as well as a recent generalization of Weyl's prescription. When $G$ is the gauge groupoid of a principal bundle…
We give a superconnection proof of Connes' index theorem for proper cocompact actions of etale groupoids. This includes Connes' general foliation index theorem for foliations with Hausdorff holonomy groupoid.
Ehresmann's introduction of differentiable groupoids in the 1950s may be seen as a starting point for two diverging lines of research, many-object Lie theory (the study of Lie algebroids and Lie groupoids) and sketch theory. This thesis uses tangent categories to build a bridge between these two lines of research, providing a structural account of Lie algebroids and the Lie functor. To accomplish this, we develop the theory of involution algebroids, which are a tangent-categorical sketch of Lie algebroids. We show that the category of Lie algebroids is precisely the category of involution alge…
In this work, we study Lie groupoids equipped with multiplicative foliations and the corresponding infinitesimal data. We determine the infinitesimal counterpart of a multiplicative foliation in terms of its core and sides together with a partial connection satisfying special properties, giving rise to the concept of IM-foliation on a Lie algebroid. The main result of this paper shows that if $G$ is a source simply connected Lie groupoid with Lie algebroid $A$, then there exists a one-to-one correspondence between multiplicative foliations on $G$ and IM-foliations on the Lie algebroid $A$.
Following Sullivan's spacial realization of a differential algebra, we construct a universal integrating Lie 2-groupoid for every Lie algebroid. Then We show that unlike Lie algebras which one-to-one correspond to simply connected Lie groups, Lie algebroids (integrable or not) one-to-one correspond to a sort of etale Lie 2-groupoids with 2-connected source fibres. Finally, we discuss how to lift Lie algebroid morphisms to Lie 2-groupoid morphisms (Lie II Theorem).
A complex Lie algebroid is a complex vector bundle over a smooth (real) manifold M with a bracket on sections and an anchor to the complexified tangent bundle of M which satisfy the usual Lie algebroid axioms. A proposal is made here to integrate analytic complex Lie algebroids by using analytic continuation to a complexification of M and integration to a holomorphic groupoid. A collection of diverse examples reveal that the holomorphic stacks presented by these groupoids tend to coincide with known objects associated to structures in complex geometry. This suggests that the object integrating…
In this paper we introduce multiplicative Dirac structures on Lie groupoids, providing a unified framework to study both multiplicative Poisson bivectors (i.e., Poisson group(oid)s) and multiplicative closed 2-forms (e.g., symplectic groupoids). We prove that for every source simply connected Lie groupoid $G$ with Lie algebroid $AG$, there exists a one-to-one correspondence between multiplicative Dirac structures on $G$ and Dirac structures on $AG$, which are compatible with both the linear and algebroid structures of $AG$. We explain in what sense this extends the integration of Lie bialgebro…
Using recently introduced Debord-Skandalis Blup's groupoids we study index theory for a compact foliated manifold with boundary inducing a foliation in its boundary. For this we consider first a blup groupoid whose Lie algebroid has sections consisting of vector fields tangent to the leaves in the interior and tangent to the leaves of the foliation in the boundary. In particular the holonomy $b$-groupoid allows us to consider the appropriate pseudodifferential calculus and the appropriate index problems. We further use the blup groupoids as the one above, and in particular its functoriality pr…
