Supermanifolds and Supergroups: Basic Theory / Edition 1by Gijs M. Tuynman
Pub. Date: 06/29/2004
Publisher: Springer Netherlands
Supermanifolds and Supergroups explains the basic ingredients of super manifolds and super Lie groups. It starts with super linear algebra and follows with a treatment of super smooth functions and the basic definition of a super manifold. When discussing the tangent bundle, integration of vector fields is treated as well as the machinery of/strong>
Supermanifolds and Supergroups explains the basic ingredients of super manifolds and super Lie groups. It starts with super linear algebra and follows with a treatment of super smooth functions and the basic definition of a super manifold. When discussing the tangent bundle, integration of vector fields is treated as well as the machinery of differential forms. For super Lie groups the standard results are shown, including the construction of a super Lie group for any super Lie algebra. The last chapter is entirely devoted to super connections.
The book requires standard undergraduate knowledge on super differential geometry and super Lie groups.
Table of Contents- Preface. - I: U-graded commutative linear algebra. 1. U-graded commutative rings and U-graded A-modules. 2. (Multi-) linear maps. 3. Direct sums, free U-graded A-modules, and quotients. 4. Tensor products. 5. Exterior powers. 6. Algebras and derivations. 7. Identifications. 8. Isomorphisms. - II: Linear algebra of free graded A-modules. 1. Our kind of Z2-graded algebra A. 2. Free graded A-modules. 3. Constructions of free graded A-modules. 4. Linear maps and matrices. 5. The graded trace and the graded determinant. 6. The body of a free graded A-module. - III: Smooth functions and A-manifolds. 1.Topology and smooth functions. 2. The structure of smooth functions. 3. Derivatives and the inverse function theorem. 4. A-manifolds. 5. Constructions of A-manifolds. - IV: Bundles. 1. Fiber bundles. 2. Constructions of fiber bundles. 3. Vector bundles and sections. 4. Constructions of vector bundles. 5. Operations on sections and on vector bundles. 6. The pull-back of a section. 7. Metrics on vector bundles. 8. Batchelor's theorem. - V: The tangent space. 1. Derivations and the tangent bundle. 2. The tangent map and some derivations. 3. Advanced properties of the tangent map. 4. Integration of vector fields. 5. Commuting flows. 6. Frobenius' theorem. 7. The exterior derivative. 8. de Rham cohomology. - VI: A-Lie groups. 1. A-Lie groups and their A-Lie algebras.2. The exponential map. 3. Convergence and the exponential of matrices. 4. Subgroups and subalgebras. 5. Homogeneous A-manifolds. 6. Pseudo effective actions. 7. Covering spaces and simply connected A-Lie groups. 8. Invariant vector fields and forms. 9. Lie's third theorem. - VII: Connections. 1. More about vector valued forms. 2. Ehresmann connections and FVF connections. 3. Connections on principal fiber bundles. 4. The exterior covariant derivative and curvature. 5. FVF connections on associated fiber bundles. 6. The covariant derivative. 7. More on covariant derivatives. 8. Forms with values in a vector bundle. 9. The covariant derivative revisited. 10. Principal fiber bundles versus sector bundles. - References. Index of Notation. Index.
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