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Complexifications of symmetric spaces and Jordan theory. (English) Zbl 0972.17018
The paper under review discusses complexifications of symmetric spaces from a geometric, resp., Jordan theoretic point of view. Since simply connected symmetric spaces are in one-to-one correspondence with Lie triple systems (the curvature tensor gives a natural trilinear operation \((x,y,z) \mapsto R(x,y)z\) on the tangent space), one obtains a natural notion of complexification by complexifying the corresponding Lie triple system and by extending the real trilinear map to a complex trilinear map. In addition to that, the author discusses the notion of a twisted (= Hermitian) complex symmetric space. Here the corresponding almost complex structure \(J\) satisfies \(R(J.x,y)z = - R(x,J.y)z\). To understand when a real symmetric space has a twisted complexification, one has to adopt a Jordan theoretic viewpoint.
According to one of the main results of the present paper, the category of symmetric spaces with a (local) twisted complexification is equivalent to the category of symmetric spaces with twist, which means that the corresponding Lie triple product \(R(x,y)z\) can be written as \(T(x,y,z) - T(y,x,z)\), where \(T\) is a Jordan triple product. The methods presented in the paper apply equally well to paracomplex symmetric spaces, i.e., where the tensor \(J\) satisfies \(J^2 = \text{id}\) instead of \(J^2 = -\text{id}\).
In the third section of the paper it is shown that the class of simple symmetric spaces decomposes into 5 classes according to their complexification behavior. This result was essentially known [S. Koh, “On affine symmetric spaces,” Trans. Am. Math. Soc. 119, 291-309 (1965; Zbl 0139.39502)], but the new proof is much shorter and more conceptual. The paper concludes with a discussion of the (twisted) {(para-)}complexifications of the simple symmetric spaces of the classical type. It is remarkable that (maybe up to central extension) every space of classical type has a twisted complexification, and it is mostly unique, but it may also happen that there are two or three of them.
The paper is a very nice contribution to the understanding of the geometric interplay between Jordan and Lie structures.

MSC:
17C36 Associated manifolds of Jordan algebras
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
22E15 General properties and structure of real Lie groups
53C35 Differential geometry of symmetric spaces
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