Projective completions of Jordan pairs. I: The generalized projective geometry of a Lie algebra. (English) Zbl 1100.17012

Summary: We prove that the projective completion \((X^+,X^-)\) of the Jordan pair \((\mathfrak g_1, \mathfrak g_{-1})\) corresponding to a 3-graded Lie algebra \(\mathfrak g=\mathfrak g_1\otimes \mathfrak g_0\otimes\mathfrak g_{-1}\) can be realized inside the space \(\mathcal F\) of inner 3-filtrations of \(\mathfrak g\) in such a way that the remoteness relation on \(X^+\times X^-\) corresponds to transversality of flags. This realization is used to give geometric proofs of structure results which will be used in Part II of this work in order to define on \(X^+\) and \(X^-\) the structure of a smooth manifold (in arbitrary dimension and over general base fields or -rings).


17C50 Jordan structures associated with other structures
17C36 Associated manifolds of Jordan algebras
17C37 Associated geometries of Jordan algebras
51A45 Incidence structures embeddable into projective geometries


Zbl 1101.17019
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