Quantization and the tangent groupoid

Abstract

This is a survey of the relationship between C*-algebraic deformation quantization and the tangent groupoid in noncommutative geometry, emphasizing the role of index theory. We first explain how C*-algebraic versions of deformation quantization are related to the bivariant E-theory of Connes and Higson. With this background, we review how Weyl--Moyal quantization may be described using the tangent groupoid. Subsequently, we explain how the Baum--Connes analytic assembly map in E-theory may be seen as an equivariant version of Weyl--Moyal quantization. Finally, we expose Connes's tangent groupoid proof of the Atiyah--Singer index theorem

16 pages, Proc. Constanta 2001

References in corpus (5)

• Deformation quantization of Poisson manifolds, I
• Operads and Motives in Deformation Quantization
• Noncommutative Manifolds the Instanton Algebra and Isospectral Deformations
• Noncommutative differential calculus, homotopy BV algebras and formality conjectures
• Lie groupoid C*-algebras and Weyl quantization

Cited by in corpus (6)

• A Schwartz type algebra for the Tangent Groupoid
• An index theorem for Wiener--Hopf operators
• Deformation quantization and the Baum-Connes conjecture
• A Continuous Field of C*-algebras and the Tangent Groupoid for Manifolds with Boundary
• Groupoids and singular manifolds
• Conceptual Differential Calculus. I: First Order Local Linear Algebra