Non-naturally reductive Einstein metrics on exceptional Lie groups

TL;DR

This paper constructs new non-naturally reductive Einstein metrics on exceptional Lie groups by analyzing fibrations over flag manifolds, solving algebraic systems with Gr"obner bases, and identifying novel metrics on ${ m G}_2$, ${ m E}_7$, and ${ m E}_8$.

Contribution

It introduces a method to find non-naturally reductive Einstein metrics on exceptional Lie groups using algebraic techniques and fibration structures, providing the first such example on ${ m G}_2$ and new metrics on ${ m E}_7$, ${ m E}_8$.

Findings

First known non-naturally reductive Einstein metric on ${ m G}_2$.

Existence of multiple non-isometric non-naturally reductive Einstein metrics on ${ m E}_7$ and ${ m E}_8$.

Application of Gr"obner bases to classify solutions of Einstein equations on 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 Gr\"obner bases and classify the real solutions of the associated algebraic systems. For the Lie group ${\rm G}_2$ we obtain the first known example of a left-invariant Einstein metric, which is not naturally reductive. Moreover, for the Lie groups ${\rm E}_7$ and ${\rm E}_8$, we conclude that there exist non-isometric non-naturally reductive Einstein metrics, which are ${\rm Ad}(K)$-invariant by different Lie subgroups $K$.