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Classification of indefinite hyper-Kaehler symmetric spaces

Dmitri V. Alekseevsky·Vicente Cortes
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Abstract

We classify indefinite simply connected hyper-Kaehler 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-Kaehler symmetric spaces of dimension 8m and orbits of the general linear group GL(m,H) over the quaternions on the space (S^4C^n)^τ of homogeneous quartic polynomials S in n = 2m complex variables satisfying the reality condition S = τS, where τis the real structure induced by the quaternionic structure of C^{2m} = H^m. We define and classify also complex hyper-Kaehler symmetric spaces. Such spaces without flat factor exist in any (complex) dimension divisible by 4.

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Dmitri V. Alekseevsky, Vicente Cortes. 2000-07-31. Classification of indefinite hyper-Kaehler symmetric spaces. https://arxiv.org/abs/math/0007189

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