The geometry of Jordan and lie structures
著者
書誌事項
The geometry of Jordan and lie structures
（Lecture notes in mathematics, 1754）
Springer, c2000
大学図書館所蔵 全83件
注記
Includes bibliographical references (p. [256]-262) and index
内容説明・目次
内容説明
0. In this work of we the Lie- and Jordan on an study interplay theory and ona level.Weintendtocontinue ittoa algebraic geometric systematicstudy ofthe role Jordan inharmonic In the of theoryplays analysis. fact, applications the of Jordan to theharmonic on cones theory algebras analysis symmetric (cf. of the wereatthe theauthor'sworkinthisarea. Then monograph[FK94]) origin Jordan in of turned the causal algebras up study many symmetric (see spaces Section and clearthat all soon itbecame XI.3), "generically" symmetric spaces have Since a relation toJordan Jordan does not significant theory. theory (yet) to the standard tools inharmonic the is text belong analysis, present designed to self-contained introduction to Jordan for readers a provide theory having basic Lie and Our ofview some on knowledge groups symmetric spaces. point is introduce first the relevant structures geometric: throughout we geometric anddeducefromtheir identities fortheassociated propertiesalgebraic algebraic structures. Thus our differs from related ones presentation (cf. e.g.
[FK94], the fact thatwe do not take an axiomatic definition ofsome [Lo77], [Sa80]) by Jordan structureasour Let us nowanoverviewof algebraic startingpoint. give the See alsothe introductions the contents. at ofeach given beginning chapter. 0.1. Lie and Jordan Ifwe the associative algebras algebras. decompose of the matrix in its and product algebra M(n,R) symmetric skew-symmetric parts, - XY YX XY YX + XY= + (0.1) 2 2 then second the term leads to the Lie with algebra gf(n,R) product [X,Y] XY- and first the termleadstotheJordan M with YX, algebra (n,R) product - X Y= + (XY YX).
目次
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
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