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Classification of indefinite hyper-Kahler symmetric spaces Dmitri V. Alekseevsky1, Vicente Cortes2

Summary: Classification of indefinite hyper-Kšahler symmetric spaces
Dmitri V. Alekseevsky1, Vicente CortŽes2
d.v.alekseevskyhull.ac.uk , vicente@math.uni-bonn.de
1 Centr "Sofus Li", Gen. Antonova 2 - 99, 117279 Moskva and Hull University, UK
Mathematisches Institut der Universitšat Bonn, Beringstr. 1, D-53115 Bonn
We classify indefinite simply connected hyper-Kšahler symmetric spaces. Any such
space without flat factor has commutative holonomy group and signature (4m, 4m).
We establish a natural 1-1 correspondence between simply connected hyper-Kšahler
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 pseudo-Riemannian 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


Source: Alekseevsky, Dmiti - Department of Mathematics, University of Hull


Collections: Computer Technologies and Information Sciences