# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
  Marcel Berger, Les espaces symétriques noncompacts, Ann. Sci. École Norm. Sup. (3) 74 (1957), 85 – 177 (French). · Zbl 0093.35602  Wolfgang Bertram, Un théorème de Liouville pour les algèbres de Jordan, Bull. Soc. Math. France 124 (1996), no. 2, 299 – 327 (French, with English and French summaries). · Zbl 0926.17020  Wolfgang Bertram, On some causal and conformal groups, J. Lie Theory 6 (1996), no. 2, 215 – 247. · Zbl 0873.17030  W. Bertram, Algebraic structures of Makarevič spaces. I, Transform. Groups 3 (1998), no. 1, 3 – 32. · Zbl 0894.22004  W. Bertram, The geometry of Jordan- and Lie structures, Habilitationsschrift (Clausthal 1999). · Zbl 1014.17024  W. Bertram, Conformal group and fundamental theorem for a class of symmetric spaces, Math. Z. 233 (2000), 39-73. CMP 2000:07  Jacques Faraut and Simon Gindikin, Pseudo-Hermitian symmetric spaces of tube type, Topics in geometry, Progr. Nonlinear Differential Equations Appl., vol. 20, Birkhäuser Boston, Boston, MA, 1996, pp. 123 – 154. · Zbl 0933.32034  N. Jacobson, General representation theory of Jordan algebras, Trans. Amer. Math. Soc. 70 (1951), 509-530. · Zbl 0044.02503  SigurÄ’ur Helgason, Differential geometry and symmetric spaces, Pure and Applied Mathematics, Vol. XII, Academic Press, New York-London, 1962.  Soji Kaneyuki and Masato Kozai, Paracomplex structures and affine symmetric spaces, Tokyo J. Math. 8 (1985), no. 1, 81 – 98. · Zbl 0585.53029  Sebastian S. Koh, On affine symmetric spaces, Trans. Amer. Math. Soc. 119 (1965), 291 – 309. · Zbl 0139.39502  Shoshichi Kobayashi and Katsumi Nomizu, Foundations of differential geometry. Vol. II, Interscience Tracts in Pure and Applied Mathematics, No. 15 Vol. II, Interscience Publishers John Wiley & Sons, Inc., New York-London-Sydney, 1969. · Zbl 0091.34802  V. Baumann, Eine parameterfreie Theorie der ungünstigsten Verteilungen für das Testen von Hypothesen, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 11 (1968), 41 – 60 (German, with English summary). · Zbl 0187.16003  Ottmar Loos, Jordan pairs, Lecture Notes in Mathematics, Vol. 460, Springer-Verlag, Berlin-New York, 1975. · Zbl 0301.17003  O. Loos, Bounded symmetric domains and Jordan pairs, Lecture Notes (Irvine 1977).  Ottmar Loos, Charakterisierung symmetrischer \?-Räume durch ihre Einheitsgitter, Math. Z. 189 (1985), no. 2, 211 – 226 (German). · Zbl 0583.53044  B.O. Makarevic, Open symmetric orbits of reductive groups in symmetric $$R$$-spaces, Math. USSR Sbornik 20 (1973), 406-418. · Zbl 0285.53041  Erhard Neher, Klassifikation der einfachen reellen speziellen Jordan-Tripelsysteme, Manuscripta Math. 31 (1980), no. 1-3, 197 – 215 (German, with English summary). · Zbl 0434.17011  Erhard Neher, Klassifikation der einfachen reellen Ausnahme-Jordan-Tripelsysteme, J. Reine Angew. Math. 322 (1981), 145 – 169 (German). · Zbl 0436.17007  Erhard Neher, On the classification of Lie and Jordan triple systems, Comm. Algebra 13 (1985), no. 12, 2615 – 2667. · Zbl 0583.17001  Ichirô Satake, Algebraic structures of symmetric domains, Kanô Memorial Lectures, vol. 4, Iwanami Shoten, Tokyo; Princeton University Press, Princeton, N.J., 1980. · Zbl 0483.32017
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.