Complexifications of symmetric spaces and Jordan theory (Q2706623)

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. NEWLINENEWLINENEWLINEAccording 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 \(J^2 = -\text{id}\). NEWLINENEWLINENEWLINEIn 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 [\textit{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. NEWLINENEWLINENEWLINEThe paper is a very nice contribution to the understanding of the geometric interplay between Jordan and Lie structures.