The Geometry of Jordan and Lie Structures

Abstract

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 XII.Tables 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