The Geometry of Jordan and Lie Structures / Edition 1

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The geometry of Jordan and Lie structures tries to answer the following question: what is the integrated, or geometric, version of real Jordan algebras, - triple systems and - pairs? Lie theory shows the way one has to go: Lie groups and symmetric spaces are the geometric version of Lie algebras and Lie triple systems. It turns out that both geometries are closely related via a functor between them, called the Jordan-Lie functor, which is constructed in this book.
The reader is not assumed to have any knowledge of Jordan theory; the text can serve as a self-contained introduction to (real finite-dimensional) Jordan theory.

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Product Details

  • ISBN-13: 9783540414261
  • Publisher: Springer Berlin Heidelberg
  • Publication date: 2/8/2001
  • Series: Lecture Notes in Mathematics Series, #1754
  • Edition description: 2000
  • Edition number: 1
  • Pages: 274
  • Product dimensions: 9.21 (w) x 6.14 (h) x 0.61 (d)

Table of Contents

First Part: The Jordan-Lie functor
I.Symetric spaces and the Lie-functor
1. Lie functor: group theoretic version
2. Lie functor:differential geometric version
3. Symmetries and group of displacements
4. The multiplication map
5. Representations os symmetric spaces
6. Examples
Appendix A: Tangent objects and their extensions
Appendix B: Affine Connections
II. Prehomogeneous symmetric spaces and Jordan algebras
1. Prehomogeneous symmetric spaces
2. Quadratic prehomogeneous symmetric spaces
3. Examples
4. Symmetric submanifolds and Helwig spaces
III. The Jordan-Lie functor
1. Complexifications of symmetric spaces
2. Twisted complex symmetric spaces and Hermitian JTS
3. Polarizations, graded Lie algebras and Jordan pairs
4. Jordan extensions and the geometric Jordan-Lie functor
IV. The classical spaces
1. Examples
2. Principles of the classification
V. Non.degenerate spaces
1. Pseudo-Riemannian symmetric spaces
2. Pseudo-Hermitian and para-Hermitian symmetric spaces
3. Pseudo-Riemannian symmetric spaces with twist
4. Semisimple Jordan algebras
5. Compact spaces and duality
Second Part: Conformal group and global theory
VI. Integration of Jordan structures
1. Circled spaces
2. Ruled spaces
3. Integrated version of Jordan triple systems
Appendix A: Integrability of almost complex structures
VII. The conformal Lie algebra
1. Euler operators and conformal Lie algebra
2. The Kantor-Koecher-Tits construction
3. General structure of the conformal Lie algebra
VIII. Conformal group and conformal completion
1. Conformal group: general properties
2. Conformal group: fine structure
3. The conformal completion and its dual
4. Conformal completion of the classical spaces
Appendix A: Some identities for Jordan triple systems
Appendix B: Equivariant bundles over homogeneous spaces
IX. Liouville theorem and fundamental theorem
1. Liouville theoremand and fundamental theorem
2. Application to the classical spaces
X. Algebraic structures of symmetric spaces with twist
1. Open symmetric orbits in the conformal completion
2. Harish-Chandra realization
3. Jordan analog of the Campbell-Hausdorff formula
4. The exponential map
5. One-parameter subspaces and Peirce-decomposition
6. Non-degenerate spaces
Appendix A: Power associativity
XI. Spaces of the first and of the second kind
1. Spaces of the first kind and Jordan algebras
2. Cayley transform and tube realizations
3. Causal symmetric spaces
4. Helwig-spaces and the extension problem
5. Examples
1. Simple Jordan algebras
2. Simple Jordan systems
3. Conformal groups and conformal completions
4. Classification of simple symmetric spaces with twist
XIII. Further topics

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