Einstein metrics on compact simple Lie groups attached to standard triples

Yan, Zaili; Deng, Shaoqiang

doi:10.1090/tran/7025

Einstein metrics on compact simple Lie groups attached to standard triples
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by Zaili Yan and Shaoqiang Deng PDF

Trans. Amer. Math. Soc. 369 (2017), 8587-8605 Request permission

Abstract:

In this paper, we study left invariant Einstein metrics on compact simple Lie groups. We find a method to construct left invariant non-naturally reductive Einstein metrics on compact simple Lie groups from a standard triple. This result, combined with the classification of standard homogeneous Einstein manifolds, leads to a large number of new Einstein metrics on compact simple Lie groups which are not naturally reductive. In particular, we show that on the compact simple Lie groups $\mathrm {SO}(8)$ and $\mathrm {SO}(10)$, there exist Einstein metrics which are not naturally reductive. A further interesting result of this paper is that on the simple Lie groups $\mathrm {SO}(2n)$ and $\mathrm {Sp}(2n)$ there exist a large number of left invariant non-naturally reductive Einstein metrics.

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