Quantization and the tangent groupoid

2002/08/02 by N.P. Landsman, N. P. Landsman, Landsman, N. P. · 2 citations
Mathematics · Physics and Astronomy · #19K65 #46L65 #Advanced Operator Algebra Research #Advanced Topics in Algebra #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Noncommutative and Quantum Gravity Theories #Operator Algebras (math.OA) #math-ph #math.MP #math.OA #msc:19K65 #msc:46L65

paper · pdf · doi:10.48550/arxiv.math-ph/0208004

16 pages, Proc. Constanta 2001

openalex publication_date 2002/08/02 · arxiv created 2002/10/16 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/29

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

Citations

• Deformations of algebras over operads and Deligne's conjecture
• Noncommutative differential calculus, homotopy BV algebras and formality conjectures
• BOTT PERIODICITY AND THE INDEX OF ELLIPTIC OPERATORS
• Twisted Lie group C^*-algebras as strict quantizations

Cited by

• An index theorem for Wiener–Hopf operators
• Deformation Quantization and the Baum–Connes Conjecture

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