A quadruple of Lie groups (G,L,K,H), where G is a compact semisimple Lie group, H ⊂ K ⊂ L are closed subgroups of G, and the related Casimir constants satisfy certain appropriate conditions, is called a basic quadruple. A basic quadruple is called Einstein if the Killing form metrics on the coset spaces G/H, G/K and G/L are all Einstein. In this paper, we first give a complete classification of the Einstein basic quadruples. We then show that, except for very few exceptions, given any quadruple (G,L,K,H) in our list, we can produce new non-naturally reductive Einstein metrics on the coset space G/H, by scaling the Killing form metrics along the complement of 𝔥 in 𝔨 and along the complement of 𝔨 in 𝔩. We also show that on some compact semisimple Lie groups, there exist a large number of left invariant non-naturally reductive Einstein metrics which are not product metrics. This discloses a new interesting phenomenon which has not been described in the literature.

中文翻译：

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正常齐次爱因斯坦流形上的非自然还原爱因斯坦度量

四倍李群(G,大号,ķ,H)， 在哪里G是紧致半单李群，H ⊂ ķ ⊂ 大号是的闭子群G，且相关的卡西米尔常数满足一定的适当条件，称为基本四元组。如果陪集空间上的 Killing 形式度量，则基本四元组称为 EinsteinG/H,G/ķ和G/大号都是爱因斯坦。在本文中，我们首先给出了爱因斯坦基本四元组的完整分类。然后我们证明，除了极少数例外，给定任何四倍(G,大号,ķ,H)在我们的列表中，我们可以在陪集空间上生成新的非自然还原爱因斯坦度量G/H，通过沿补码缩放 Killing 形式指标𝔥在𝔨并沿补𝔨在𝔩. 我们还表明，在一些紧凑的半单李群上，存在大量不是乘积度量的左不变非自然还原爱因斯坦度量。这揭示了一种新的有趣现象，该现象尚未在文献中描述。