Deformations of Lie Groupoids

2015/10/31 by Marius Crainic, João Nuno Mestre, Ivan Struchiner · 26 citations
Mathematics · #Adjoint representation #Advanced Topics in Algebra #Algebra over a field #Algebraic structures and combinatorial models #Cohomology #Homotopy and Cohomology in Algebraic Topology #Lie group #Mathematics #Pure mathematics #Rigidity (electromagnetism) #math.DG #msc:58H05 #msc:58H15

paper · pdf · doi:10.1093/imrn/rny221

published in International Mathematics Research Notices 2020(21), 7662-7746 (Oxford University Press) · 45 pages. Published version; corrected typos

openalex publication_date 2018/08/30 · arxiv created 2020/11/17 · arxiv updated 2020/11/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06

Abstract

Abstract We study deformations of Lie groupoids by means of the cohomology which controls them. This cohomology turns out to provide an intrinsic model for the cohomology of a Lie groupoid with values in its adjoint representation. We prove several fundamental properties of the deformation cohomology including Morita invariance, a van Est theorem, and a vanishing result in the proper case. Combined with Moser’s deformation arguments for groupoids, we obtain several rigidity and normal form results.

Citations

• Complex Manifolds and Deformation of Complex Structures
• Introduction to Foliations and Lie Groupoids
• Stability and invariants of Hilsum-Skandalis maps
• Differentiable and algebroid cohomology, Van Est isomorphisms, and characteristic classes
• The Cohomology of Differentiable Transformation Groups