Abstract.
Following our approach to metric Lie algebras developed in a previous paper we propose a way of understanding pseudo-Riemannian symmetric spaces which are not semisimple. We introduce cohomology sets (called quadratic cohomology) associated with orthogonal modules of Lie algebras with involution. Then we construct a functorial assignment which sends a pseudo-Riemannian symmetric space M to a triple consisting of:
-
(i) a Lie algebra with involution (of dimension much smaller than the dimension of the transvection group of M);
-
(ii) a semisimple orthogonal module of the Lie algebra with involution; and
-
(iii) a quadratic cohomology class of this module.
That leads to a classification scheme of indecomposable nonsimple pseudo-Riemannian symmetric spaces. In addition, we obtain a full classification of symmetric spaces of index 2 (thereby completing and correcting in part earlier classification results due to Cahen and Parker and to Neukirchner).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
D. V. Alekseevsky, V. Cortés, Classification of pseudo-Riemannian symmetric spaces of quaternionic Kähler type, in: Lie Groups and Invariant Theory, Amer. Math. Soc. Transl. Ser. 2, Vol. 213, Providence, RI, 2005, PP. 33–62.
B. B. Aстраханцев, Сuммеmpuческuче npосmpaнcmвa корaн Ϩ a 1, Mат. Cб. 96(138) (1975), 135–151. Engl. transl.: V. V. Astrakhantsev, Symmetric spaces of corank 1, Math. USSR, Sb. 25 (1975), (1976), 129–144.
M. Berger, Les espaces symétriques non compacts, Ann. Sci. École Norm. Sup. 74 (1957), 85–177.
L. Bérard-Bergery, Décomposition de Jordan–Hölder d’une représentation de dimension finie, adaptée à une forme réexive, handwritten notes.
L. Bérard-Bergery, Structure des espaces symétriques pseudo-Riemanniens, handwritten notes.
M. Blau, J. M. Figueroa-O’Farrill, G. Papadopoulos, Penrose limits, supergravity and brane dynamics, Classical Quantum Gravity 19 (2002), 4753–4805.
M. Cahen, M. Parker, Sur des classes d’espaces pseudo-Riemanniens symétriques, Bull. Soc. Math. Belg. 22 (1970), 339–354.
M. Cahen, M. Parker, Pseudo-Riemannian Symmetric Spaces, Mem. Amer. Math. Soc. 24 (1980), no. 229.
M. Cahen, N.Wallach, Lorentzian symmetric spaces, Bull. Amer. Math. Soc. 76 (1970), 585–591.
V. Cortés, Odd Riemannian symmetric spaces associated to four-forms, Math. Scand. 98 (2006), 201–216.
S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, Graduate Studies in Mathematics, Vol. 34, Amer. Math. Soc., Providence, RI, 2001.
I. Kath, Indefinite extrinsic symmetric spaces I, arXiv:0809.4713.
I. Kath, M. Olbrich, Metric Lie algebras and quadratic extensions, Transform. Groups 11 (2006), 87–131.
I. Kath, M. Olbrich, New examples of hyper-Kähler symmetric spaces, J. Geom. Phys. 53 (2007), no. 3, 345–353.
I. Kath, M. Olbrich, The classification problem for pseudo-Riemannian symmetric spaces, in: Recent developments in pseudo-Riemannian Geometry (eds. D. V. Alekseevsky, H. Baum), ESI Lectures in Mathematics and Physics, Eur. Math. Soc., Zürich, 2008, pp. 1–52.
S. Kobayashi, K. Nomizu, Foundations of Differential Geometry, Vol. II, Reprint of the 1969 original, Wiley Classics Library, Wiley, New York, 1996.
Th. Neukirchner, Solvable pseudo-Riemannian symmetric spaces, arXiv:math.DG/0301326.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Kath, I., Olbrich, M. On the structure of pseudo-Riemannian symmetric spaces. Transformation Groups 14, 847–885 (2009). https://doi.org/10.1007/s00031-009-9071-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00031-009-9071-z