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@plain_simon You might also be interested in work of Wolfgang Bertram, specifically https://t.co/MagFse0yJ0 He's not into loops but thinks a lot about integration of Jordan structures.
The Geometry of Jordan and Lie Structures
Overview of attention for book
Table of Contents
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Altmetric BadgeBook Overview
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Altmetric BadgeChapter 1 Chapter I: Symmetric spaces and the Lie-functor
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Altmetric BadgeChapter 2 Chapter II: Prehomogeneous symmetric spaces and Jordan algebras
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Altmetric BadgeChapter 3 Chapter III: The Jordan-Lie functor
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Altmetric BadgeChapter 4 Chapter IV: The classical spaces
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Altmetric BadgeChapter 5 Chapter V: Non-degenerate spaces
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Altmetric BadgeChapter 6 Chapter VI: Integration of Jordan structures
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Altmetric BadgeChapter 7 Chapter VII: The conformal Lie algebra
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Altmetric BadgeChapter 8 Chapter VIII: Conformal group and conformal completion
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Altmetric BadgeChapter 9 Chapter IX: Liouville theorem and fundamental theorem
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Altmetric BadgeChapter 10 Chapter X: Algebraic structures of symmetric spaces with twist
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Altmetric BadgeChapter 11 Chapter XI: Spaces of the first and of the second kind
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Altmetric BadgeChapter 12 Chapter XII: Tables
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Altmetric BadgeChapter 13 Chapter XIII: Further topics