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For every \({\mathbb{Z}}_{2}\) -graded Clifford module we construct a pair of homogeneous convex cones based on Vinberg’s theory. We show that one member of the pair is always generated by a cubic homogeneous convex hypersurface. By results of de Wit and Van Proeyen, it is known that such hypersurfaces give rise to homogeneous quaternionic Kähler manifolds of negative scalar curvature. If the cone generated by the cubic hypersurface is self-dual, then the corresponding quaternionic Kähler manifold is a symmetric spaces of non-compact type dual to a Wolf space (i.e. to quaternionic Kähler symmetric spaces of compact type).

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Homogeneous Cones Associated with Clifford Modules

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  • 发表日期：

    2025年

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  • D. V. Alekseevsky,
  • V. Cortés

摘要

For every \({\mathbb{Z}}_{2}\) -graded Clifford module we construct a pair of homogeneous convex cones based on Vinberg’s theory. We show that one member of the pair is always generated by a cubic homogeneous convex hypersurface. By results of de Wit and Van Proeyen, it is known that such hypersurfaces give rise to homogeneous quaternionic Kähler manifolds of negative scalar curvature. If the cone generated by the cubic hypersurface is self-dual, then the corresponding quaternionic Kähler manifold is a symmetric spaces of non-compact type dual to a Wolf space (i.e. to quaternionic Kähler symmetric spaces of compact type).