Quantization of Poisson algebras associated to Lie algebroids

2000/01/04 by N. P. Landsman, Landsman, N. P., B. Ramazan +1 · 3 citations
Mathematics · Physics and Astronomy · #46L65 #81S99 #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Operator Algebras (math.OA) #Quantum Algebra (math.QA) #Symplectic Geometry (math.SG) #math-ph #math.MP #math.OA #math.QA #math.SG #msc:46L65 #msc:81S99

paper · pdf · doi:10.48550/arxiv.math-ph/0001005

34 pages, minor corrections

arxiv created 2000/11/21 · arxiv updated 2009/11/30

Abstract

We prove the existence of a strict deformation quantization for the canonical Poisson structure on the dual of an integrable Lie algebroid. It follows that any Lie groupoid C*-algebra may be regarded as a result of a quantization procedure. The C*-algebra of the tangent groupoid of a given Lie groupoid G (with Lie algebra g) is the C*-algebra of a continuous field of C*-algebras over R with fibers A₀=C*(g)=C₀(g*) and A_h=C*(G) for nonzero h. The same is true for the corresponding reduced C*-algebras. Our results have applications to, e.g., transformation group C*-algebras, K-theory, and index theory.

Cited by

• A Volterra Calculus for Lie Groupoids
• Spectral continuity for étale groupoids with the Rapid decay property
• Cotangent Bundles as Coadjoint Orbits and Asymptotic Character Formulas

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