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The Geometry of Jordan and Lie Structures : Lecture Notes in Mathematics - Wolfgang Bertram

The Geometry of Jordan and Lie Structures

Lecture Notes in Mathematics

Paperback Published: February 2001
ISBN: 9783540414261
Number Of Pages: 274

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First Part: The Jordan-Lie functorI.Symetric spaces and the Lie-functor1. Lie functor: group theoretic version2. Lie functor:differential geometric version3. Symmetries and group of displacements4. The multiplication map5. Representations os symmetric spaces6. ExamplesAppendix A: Tangent objects and their extensionsAppendix B: Affine ConnectionsII. Prehomogeneous symmetric spaces and Jordan algebras1. Prehomogeneous symmetric spaces2. Quadratic prehomogeneous symmetric spaces3. Examples4. Symmetric submanifolds and Helwig spacesIII. The Jordan-Lie functor1. Complexifications of symmetric spaces2. Twisted complex symmetric spaces and Hermitian JTS3. Polarizations, graded Lie algebras and Jordan pairs4. Jordan extensions and the geometric Jordan-Lie functorIV. The classical spaces1. Examples2. Principles of the classificationV. Non.degenerate spaces1. Pseudo-Riemannian symmetric spaces2. Pseudo-Hermitian and para-Hermitian symmetric spaces3. Pseudo-Riemannian symmetric spaces with twist4. Semisimple Jordan algebras5. Compact spaces and dualitySecond Part: Conformal group and global theoryVI. Integration of Jordan structures1. Circled spaces2. Ruled spaces3. Integrated version of Jordan triple systemsAppendix A: Integrability of almost complex structuresVII. The conformal Lie algebra1. Euler operators and conformal Lie algebra2. The Kantor-Koecher-Tits construction3. General structure of the conformal Lie algebraVIII. Conformal group and conformal completion1. Conformal group: general properties2. Conformal group: fine structure3. The conformal completion and its dual4. Conformal completion of the classical spacesAppendix A: Some identities for Jordan triple systemsAppendix B: Equivariant bundles over homogeneous spacesIX. Liouville theorem and fundamental theorem1. Liouville theorem and and fundamental theorem2. Application to the classical spacesX. Algebraic structures of symmetric spaces with twist1. Open symmetric orbits in the conformal completion2. Harish-Chandra realization3. Jordan analog of the Campbell-Hausdorff formula4. The exponential map5. One-parameter subspaces and Peirce-decomposition6. Non-degenerate spacesAppendix A: Power associativity XI. Spaces of the first and of the second kind1. Spaces of the first kind and Jordan algebras2. Cayley transform and tube realizations3. Causal symmetric spaces4. Helwig-spaces and the extension problem5. ExamplesXII.Tables1. Simple Jordan algebras2. Simple Jordan systems3. Conformal groups and conformal completions4. Classification of simple symmetric spaces with twistXIII. Further topics

Introduction
Symmetric spaces and the Lie-functorp. 1
Lie functor: group theoretic versionp. 2
Lie functor: differential geometric versionp. 6
Symmetries and group of displacementsp. 10
The multiplication mapp. 13
Representations of symmetric spacesp. 15
Examplesp. 18
Tangent objects and their extensionsp. 32
Affine Connectionsp. 35
Prehomogeneous symmetric spaces and Jordan algebrasp. 42
Prehomogeneous symmetric spacesp. 42
Quadratic prehomogeneous symmetric spacesp. 45
Examplesp. 54
Symmetric submanifolds and Helwig spacesp. 57
The Jordan-Lie functorp. 61
Complexifications of symmetric spacesp. 62
Twisted complex symmetric spaces and Hermitian JTSp. 65
Polarizations, graded Lie algebras and Jordan pairsp. 70
Jordan extensions and the geometric Jordan-Lie functorp. 74
The Classical spacesp. 81
Examplesp. 82
Principles of classificationp. 92
Non-degenerate spacesp. 97
Pseudo-Riemannian symmetric spacesp. 98
Pseudo-Hermitian and para-Hermitian symmetric spacesp. 103
Pseudo-Riemannian symmetric spaces with twistp. 106
Semisimple Jordan algebrasp. 107
Compact spaces and dualityp. 111
Integration of Jordan structuresp. 116
Circled spacesp. 118
Ruled spacesp. 120
Integrated version of Jordan triple systemsp. 122
Integrability of almost complex structuresp. 124
The conformal Lie algebrap. 127
Euler operators and conformal Lie algebrap. 128
The Kantor-Koecher-Tits constructionp. 132
General structure of the conformal Lie algebrap. 138
Conformal group and conformal completionp. 143
Conformal group: general propertiesp. 144
Conformal group: fine structurep. 150
The conformal completion and its dualp. 156
Conformal completion of the classical spacesp. 160
Some identities for Jordan triple systemsp. 166
Equivariant bundles over homogeneous spacesp. 167
Liouville theorem and fundamental theoremp. 171
Liouville theorem and fundamental theoremp. 171
Application to the classical spacesp. 177
Algebraic structures of symmetric spaces with twistp. 184
Open symmetric orbits in the conformal completionp. 185
Harish-Chandra realizationp. 188
Jordan analog of the Campbell-Hausdorff formulap. 192
The exponential mapp. 200
One-parameter subspaces and Peirce-decompositionp. 204
Non-degenerate spacesp. 208
Power associativityp. 213
Spaces of the first and of the second kindp. 216
Spaces of the first kind and Jordan algebrasp. 216
Cayley transform and tube realizationsp. 220
Causal symmetric spacesp. 226
Helwig-spaces and the extension problemp. 230
Examplesp. 232
Tablesp. 240
Simple Jordan algebrasp. 240
Simple Jordan triple systemsp. 243
Conformal groups and conformal completionsp. 245
Classification of simple symmetric spaces with twistp. 248
Further topicsp. 254
Bibliographyp. 256
Notationp. 263
Indexp. 266
Table of Contents provided by Blackwell. All Rights Reserved.

ISBN: 9783540414261
ISBN-10: 3540414266
Series: Lecture Notes in Mathematics
Audience: General
Format: Paperback
Language: English
Number Of Pages: 274
Published: February 2001
Publisher: Springer-Verlag Berlin and Heidelberg Gmbh & Co. Kg
Country of Publication: DE
Dimensions (cm): 23.39 x 15.6  x 1.55
Weight (kg): 0.41