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 ${\mathrm{G}}_2$ we obtain the first known example of a left-invariant Einstein metric, which is not naturally reductive. Moreover, for the Lie groups ${\mathrm{E}}_7$ and ${\mathrm{E}}_8$, we conclude that there exist non-isometric non-naturally reductive Einstein metrics, which are $\mathrm{Ad}(K)$-invariant by different Lie subgroups $K$.

33 pages

References in corpus (1)

• The Ricci flow approach to homogeneous Einstein metrics on flag manifolds

Cited by in corpus (6)

• On left-invariant Einstein Riemannian metrics that are not geodesic orbit
• Notes on "Einstein metrics on compact simple Lie groups attached to standard triples"
• Einstein Lie groups, geodesic orbit manifolds and regular Lie subgroups
• New Non-naturally reductive Einstein metrics on Exceptional simple Lie groups
• Homogeneous Einstein metrics on non-Kähler C-spaces
• New Einstein metrics on the Lie group $SO(n)$ which are not naturally reductive