Non-naturally reductive Einstein metrics on exceptional Lie groups

作者：Chrysikos Ioannis^*; Sakane Yusuke

来源：Journal of Geometry and Physics, 2017, 116: 152-186.

DOI：10.1016/j.geomphys.2017.01.030

• Left-invariant Einstein metrics
• Naturally reductive metrics
• Exceptional Lie groups
• Flag manifolds

Given an exceptional compact simple Lie group G we describe new left-invariant Einstein metrics which are not naturally reductive. In particular, we consider fibrations of G over flag manifolds with a certain kind of isotropy representation and we construct the Einstein equation with respect to the induced left-invariant metrics. Then we apply a technique based on Grobner bases and classify the real solutions of the associated algebraic systems. For the Lie group G(2) we obtain the first known example of a left-invariant Einstein metric, which is not naturally reductive. Moreover, for the Lie groups E-7 and E-8, we conclude that there exist non-isometric non-naturally reductive Einstein metrics, which are Ad(K)-invariant by different Lie subgroups K.

• 出版日期2017-6