Supermanifolds and Supergroups: Basic Theory

ISBN-13: 9789048166329 / Angielski / Miękka / 2010 / 416 str.

Gijs M. Tuynman

Opis

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.

Recenzje

From the reviews:

"The author provides an introduction to the theory of supermanifolds ... . The book is written in a logical order and it is self-contained. ... The list of references indicates also where in the book the reference is quoted. Therefore, I find this book very useful, a true instrument of reference and study. It can be read by graduate students in mathematics and physics and by anyone interested in this subject and it presents in very pleasant and accessible manner all the basics ... ." (Eugen Pascu, Zentralblatt MATH, Vol. 1083 (9), 2006)

"This book is an introduction to super differential geometry at the level of graduate studies in mathematics and theoretical physics. ... the book presents in a clear and precise way difficult notions and theories about supermanifolds and supergroups. It will be easily readable by the people for whom it is intended, i.e. graduate students in mathematics and theoretical physics ... . It is self-contained ... . the book will be very useful for both beginners and active researchers in the field of supermanifolds and supergroups." (Anargyros Fellouris, Mathematical Reviews, Issue 2005 k)

Spis treści

- 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.