2011/12/20 by Markus J. Pflaum, M. J. Pflaum, Hessel Posthuma +6
Mathematics · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Differential Geometry (math.DG) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #K-Theory and Homology (math.KT) #Quantum Algebra (math.QA) #math.DG #math.KT #math.QA
paper · pdf · doi:10.48550/arxiv.1112.4857
40 pages
arxiv created 2011/12/20 · openalex publication_date 2011/12/20 · arxiv updated 2011/12/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We define the "localized index" of longitudinal elliptic operators on Lie groupoids associated to Lie algebroid cohomology classes. We derive a topological expression for these numbers using the algebraic index theorem for Poisson manifolds on the dual of the Lie algebroid. Underlying the definition and computation of the localized index, is an action of the Hopf algebroid of jets around the unit space, and the characteristic map it induces on Lie algebroid cohomology. This map can be globalized to differentiable groupoid cohomology, giving a definition as well as a computation of the "global index". The correspondence between the "global" and "localized" index is given by the van Est map for Lie groupoids.