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Homogeneous Einstein manifolds based on symplectic triple systems. (English) Zbl 1470.53053

In the paper under review, the author proposes a purely algebraic construction that gives reductive pair related to a semi-Riemannian homogeneous manifold from a given simple symplectic triple system over the real numbers, the standard enveloping Lie algebra and the algebra of inner derivations of the triple. Moreover, the obtained semi-Riemannian manifold is Einstein. The idea of the corresponding construction is inspired by similar considerations in the case of 3-Sasakian homogeneous manifolds [C. Draper et al., Math. Z. 294, No. 1-2, 817–868 (2020; 07152531)].
As the author himself notes, the corresponding Einstein semi-Riemannian homogeneous manifold can be obtained also as the total spaces of principal bundles of standard basis on para-quaternionic Kähler symmetric spaces [D. Alekseevsky and V. Cortés, Osaka J. Math. 45, No. 1, 215–251 (2008; 1177.53047)].

MSC:

53C30 Differential geometry of homogeneous manifolds
53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
17A40 Ternary compositions
17B60 Lie (super)algebras associated with other structures (associative, Jordan, etc.)
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