General Theory of Lie Groupoids and Lie Algebroids

2005/06/09 by Kirill C. H. Mackenzie · 26 citations
Mathematics · #Homotopy and Cohomology in Algebraic Topology #Advanced Topics in Algebra #Algebraic structures and combinatorial models

paper · doi:10.1017/cbo9781107325883

openalex publication_date 2005/06/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/04

Abstract

This a comprehensive modern account of the theory of Lie groupoids and Lie algebroids, and their importance in differential geometry, in particular their relations with Poisson geometry and general connection theory. It covers much work done since the mid 1980s including the first treatment in book form of Poisson groupoids, Lie bialgebroids and double vector bundles, as well as a revised account of the relations between locally trivial Lie groupoids, Atiyah sequences, and connections in principal bundles. As such, this book will be of great interest to all those concerned with the use of Poisson geometry as a semi-classical limit of quantum geometry, as well as to all those working in or wishing to learn the modern theory of Lie groupoids and Lie algebroids.

Citations

• Lie Groupoids and Lie Algebroids in Differential Geometry
• Integrability of Lie brackets

Cited by

• GEODESICS ON RIEMANNIAN STACKS
• Lagrangian Mechanics and Reduction on Fibered Manifolds
• Affine deformations of cotangent groupoids
• Equivariant Cohomology over Lie Groupoids and Lie–Rinehart Algebras
• Foliated groupoids and infinitesimal ideal systems
• Symplectic algebroids, groupoid Toeplitz operators and deformation quantization
• Associative half-densities on symplectic groupoids and quantization
• A unified framework for mechanics: Hamilton–Jacobi equation and applications
• Parametrized L-infinity Spaces
• An Explicit Construction of \mathbbS¹-Gerbes over the Stack [G/G]
• Obstruction classes of crossed modules of Lie algebroids and Lie groupoids linked to existence of principal bundles
• Poisson Manifolds, Lie Algebroids, Modular Classes: a Survey
• Algebraic Topology Foundations of Supersymmetry and Symmetry Breaking in Quantum Field Theory and Quantum Gravity: A Review
• Quantum Systems as Lie Algebroids
• Lie algebroid fibrations
• On the integration of Poisson homogeneous spaces
• Fiber-wise linear Poisson structures related to W∗-algebras
• Jacobi bundles and the BFV-complex
• Twisted Differential String and Fivebrane Structures
• A Hodge-type decomposition of holomorphic Poisson cohomology on nilmanifolds
• Nontrivial constitutive laws and unified structures in constrained BF theory
• Higher vector bundles and multi-graded symplectic manifolds
• Almost Kähler Ricci Flows and Einstein and Lagrange–Finsler Structures on Lie Algebroids
• Shifted Derived Poisson Manifolds Associated with Lie Pairs
• Higher-Order Analogs of Lie Algebroids via Vector Bundle Comorphisms
• Lie algebroid [wikipedia]