Classification of indefinite hyper-Kaehler symmetric spaces

2000/07/31 by Dmitri V. Alekseevsky, Vicente Cortés, Alekseevsky, Dmitri V. +2
Mathematics · Physics and Astronomy · #53C25 #53C35 #Advanced Differential Geometry Research #Algebraic and Geometric Analysis #Differential Geometry (math.DG) #FOS: Mathematics #Geometry and complex manifolds #math.DG #msc:53C25 #msc:53C35

paper · pdf · doi:10.48550/arxiv.math/0007189

28 pages, Latex

arxiv created 2000/07/31 · openalex publication_date 2000/07/31 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 (S⁴Cⁿ)^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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