Title Lie groupoids and Lie algebroids in differential geometry / K. Mackenzie

Published Cambridge [Cambridgeshire] ; New York : Cambridge University Press, 1987

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Description 1 online resource (xvi, 327 pages) : illustrations Series London Mathematical Society lecture note series ; 124 London Mathematical Society lecture note series ; 124. Contents Cover; Title; Copyright; Contents; Introduction; Dedication; CHAPTER I The algebra of groupoids; Introduction; 1. Groupoids; 2. Morphisms, subgroupoids and quotient groupoids; 3. Transitive and totally intransitive groupoids; CHAPTER II Topological groupoids; Introduction; 1. Basic definitions and examples; 2. Local triviality; 3. Components in topological groupoids; 4. Representations of topological groupoids; 5. Admissible sections; 6. The monodromy groupoid of a locally trivial groupoid; 7. Path connections in topological groupoids; CHAPTER III Lie groupoids and Lie algebroids Introduction1. Differentiable and Lie groupoids; 2. Lie algebroids; 3. The Lie algebroid of a differentiable groupoid; 4. The exponential map and adjoint formulas; 5. Infinitesimal connection theory and the concept of transition form; 6. The Lie theory of Lie groupoids over a fixed base; 7. Path connections in Lie groupoids; CHAPTER IV The cohomology of Lie algebroids; Introduction; 1. The abstract theory of transitive Lie algebroids; 2. The cohomology of Lie algebroids; 3. Non-abelian extensions of Lie algebroids and the existence of transitive Lie algebroids with prescribed curvature 4. The existence of local flat connections and families of transition forms5. The spectral sequence of a transitive Lie algebroid; CHAPTER V An obstruction to the Integrability of transitive Lie algebroids; Introduction; 1. Results; 2. Epilogue; APPENDIX A On principal bundles and Atiyah sequences; Introduction; 1. Principal and fibre bundles; 2. Quotients of vector bundles over group actions; 3. The Atiyah sequence of a principal bundle; 4. Infinitesimal connections and curvature; APPENDIX B On Lie groups and Lie algebras; Introduction; 1. Definitions and notations 2. Formulas for the right derivativeAPPENDIX C On vector bundles; REFERENCES; INDEX Summary This book provides a striking synthesis of the standard theory of connections in principal bundles and the Lie theory of Lie groupoids. The concept of Lie groupoid is a little-known formulation of the concept of principal bundle and corresponding to the Lie algebra of a Lie group is the concept of Lie algebroid: in principal bundle terms this is the Atiyah sequence. The author's viewpoint is that certain deep problems in connection theory are best addressed by groupoid and Lie algebroid methods. After preliminary chapters on topological groupoids, the author gives the first unified and detailed account of the theory of Lie groupoids and Lie algebroids. He then applies this theory to the cohomology of Lie algebroids, re-interpreting connection theory in cohomological terms, and giving criteria for the existence of (not necessarily Riemannian) connections with prescribed curvature form. This material, presented in the last two chapters, is work of the author published here for the first time. This book will be of interest to differential geometers working in general connection theory and to researchers in theoretical physics and other fields who make use of connection theory Bibliography Includes bibliographical references (pages 317-322) and index Notes Print version record Subject Connections (Mathematics) Lie groupoids. Lie algebroids. Fiber bundles (Mathematics) Lie groups. MATHEMATICS -- Geometry -- Differential. Lie groups Connections (Mathematics) Fiber bundles (Mathematics) Lie algebroids Lie groupoids Lie-Gruppoid Lie-Algebroid Differentialgeometrie Gruppoid Lie-Algebra Faisceaux fibrés (mathématiques) Endomorphismes (théorie des groupes) Lie, Groupes de. Form Electronic book LC no. 87010287 ISBN 9781107361454 1107361451 9780511661839 0511661835

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