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Einstein metrics on compact simple Lie groups attached to standard triples


Authors: Zaili Yan and Shaoqiang Deng
Journal: Trans. Amer. Math. Soc. 369 (2017), 8587-8605
MSC (2010): Primary 53C25, 53C35, 53C30
DOI: https://doi.org/10.1090/tran/7025
Published electronically: May 5, 2017
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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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Additional Information

Zaili Yan
Affiliation: Department of Mathematics, Ningbo University, Ningbo, Zhejiang Province, 315211, People’s Republic of China
Email: yanzaili@nbu.edu.cn

Shaoqiang Deng
Affiliation: School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, People’s Republic of China
Email: dengsq@nankai.edu.cn

DOI: https://doi.org/10.1090/tran/7025
Keywords: Einstein metrics, compact Lie groups, naturally reductive metrics, standard homogeneous Einstein manifolds
Received by editor(s): October 15, 2015
Received by editor(s) in revised form: January 27, 2016
Published electronically: May 5, 2017
Additional Notes: The first author was supported by NSFC (Nos. 11626134, 11401425) and K. C. Wong Magna Fund in Ningbo University.
The second author was supported by NSFC (Nos. 11671212, 51535008) of China
Article copyright: © Copyright 2017 American Mathematical Society