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Jordan- and Lie geometries. (English) Zbl 1313.17034

Summary: In these lecture notes we report on research aiming at understanding the relation between algebras and geometries, by focusing on the classes of Jordan algebraic and of associative structures and comparing them with Lie structures. The geometric object sought for, called a generalized projective, resp. an associative geometry, can be seen as a combination of the structure of a symmetric space, resp. of a Lie group, with the one of a projective geometry. The text is designed for readers having basic knowledge of Lie theory – we give complete definitions and explain the results by presenting examples, such as Grassmannian geometries.

MSC:

17C37 Associated geometries of Jordan algebras
20N10 Ternary systems (heaps, semiheaps, heapoids, etc.)
22A30 Other topological algebraic systems and their representations
51B25 Lie geometries in nonlinear incidence geometry
51P05 Classical or axiomatic geometry and physics
16W10 Rings with involution; Lie, Jordan and other nonassociative structures
81P05 General and philosophical questions in quantum theory
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