 
Summary: Classification of indefinite hyperKšahler symmetric spaces
Dmitri V. Alekseevsky1, Vicente CortŽes2
d.v.alekseevskyhull.ac.uk , vicente@math.unibonn.de
1 Centr "Sofus Li", Gen. Antonova 2  99, 117279 Moskva and Hull University, UK
2
Mathematisches Institut der Universitšat Bonn, Beringstr. 1, D53115 Bonn
Abstract
We classify indefinite simply connected hyperKšahler symmetric spaces. Any such
space without flat factor has commutative holonomy group and signature (4m, 4m).
We establish a natural 11 correspondence between simply connected hyperKšahler
symmetric spaces of dimension 8m and orbits of the group GL(m, H) on the space
(S4Cn) of homogeneous quartic polynomials S in n = 2m complex variables sat
isfying the reality condition S = S, where is the real structure induced by the
quaternionic structure of C2m = Hm. We define and classify also complex hyper
Kšahler symmetric spaces. Such spaces without flat factor exist in any (complex)
dimension divisible by 4.
1 Introduction
We recall that a pseudoRiemannian manifold (M, g) is called a symmetric space if any point
x M is an isolated fixed point of an involutive isometry sx (called central symmetry with
centre x). Since the product of two central symmetries sx and sy with sufficiently close centres
