The Geometry of Jordan and Lie Structures (eBook, PDF)
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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 finit...
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. TOC:First Part: The Jordan-Lie functor. Symmetric spaces and the Lie-functor; Prehomogeneous symmetric spaces and Jordan algebras; The Jordan-Lie functor, The classical spaces; Non-degenerate spaces. Second Part: Conformal group and global theory. Integration of the Jordan structures; The conformal Lie algebra; Conformal group and conformal completion; Liouville theorem and fundamental theorem; Algebraic structures of symmetric spaces with twist; Spaces of the first and of the second kind; Tables; Further topics.
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