Non-naturally reductive Einstein metrics on exceptional Lie groups

Abstract

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 Gröbner 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 .