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By J.N. Coldstream

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Note that the values of d and dH for N D R2 in the table are not correct for small n, namely for those n for which the value of d given in the table would be negative. We do not consider these special cases in the table. In order to not present only vague data here let us study one case in more detail. 1/ D fŒX; Y  D Zg. 1/2 modul a. l; a/. l; Â l /-module for Â l given by lC D R Z; l D spanfX; Y g. e e 28 I. Kath and M. Olbrich Table 1. 2; n/ l; An l;Â l N R1 R 2 lDl DR 1. 1;p ; ap;n C 2. 2;p ; ap;n C 3.

D !. /, and H ˝C E. We denote all The classiﬁcation problem for pseudo-Riemannian symmetric spaces 37 these structures by the same symbol . H ˝C E/ carries a canonical structure of a hyper-Kähler symmetric triple. hS / . This construction produces all hyper-Kähler symmetric triples. 9 (Alekseevsky–Cortés [2]). g; ˆ; h ; i/ Š gJ;S as a hyper-Kähler symmetric triple. g; ˆ; h ; i/ up to complex linear isomorphisms. Thus the classification of hyper-Kähler symmetric triples is equivalent to the classification of all -invariant solutions of (10).

The idea behind it is very simple. Let G be the transvection group of M , and let J G be the analytic subgroup corresponding to the ideal i? g. Then J acts on M , and we would like to set q to be the projection onto the orbit space N D J nM . That the orbits are flat and coisotropic is a simple consequence of the properties of i? The main problem is to show that the orbit space is a manifold (the orbits have to be closed, in particular). It is this point, where we need Condition (a) or (b). Without these conditions, it is not difficult to construct examples with non-closed J -orbits.