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Jordan structures in geometry and physics. With an appendix on Jordan structures in analysis. (English) Zbl 1073.17014

Bucharest: Editura Academiei Române (ISBN 973-27-0956-1/pbk). 201 p. (2003).
This book provides an impressive broad survey of the use of Jordan structures (Jordan algebras and Jordan triples) in other branches of mathematics, especially in geometry, and in physics. It contains “only” definitions, results and comments, including open problems and many historical comments. For details and proofs, the reader is referred to an extensive bibliography of 847 items.
The following areas are covered:
- Jordan structures in differential geometry: symmetric cones, Hermitian symmetric spaces, some Riemannian symmetric spaces, causal structures, infinite-dimensional Grassmannians, Kadomtsev-Petviashvili equation;
- Jordan algebras in ring geometries, such as octonion planes (Moufang projective plane over the Cayley algebra), Barbilian geometries;
- Jordan structures in physics: quantum mechanics, quantum group theory, string theory.
An appendix is devoted to Jordan structures in analysis: one part is on Jordan-Banach structures; the second part of this appendix gives a few indications on analysis problems in symmetric spaces: Hua equations, kernel functions. This part could be a chapter by itself, as important as the three main chapters of the book: Jordan structures in harmonic analysis and representation theory.

MSC:

17C50 Jordan structures associated with other structures
17-02 Research exposition (monographs, survey articles) pertaining to nonassociative rings and algebras
17C65 Jordan structures on Banach spaces and algebras
32M15 Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects)
35Q58 Other completely integrable PDE (MSC2000)
51A35 Non-Desarguesian affine and projective planes
53C35 Differential geometry of symmetric spaces
46H70 Nonassociative topological algebras
46K70 Nonassociative topological algebras with an involution
81R12 Groups and algebras in quantum theory and relations with integrable systems
81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
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