New Einstein metrics on the Lie group $SO(n)$ which are not naturally reductive

TL;DR

This paper discovers new invariant Einstein metrics on the compact Lie groups SO(n) for n ≥ 7 that are not naturally reductive, using symmetry assumptions and symbolic polynomial computations.

Contribution

It introduces a novel method to find non-naturally reductive Einstein metrics on SO(n) by applying symmetry constraints and Gr"obner basis computations.

Findings

Identified new Einstein metrics on SO(n) for n ≥ 7

Demonstrated metrics are not naturally reductive

Used symbolic computation to solve polynomial systems

Abstract

We obtain new invariant Einstein metrics on the compact Lie groups $SO(n)$ ($n \geq 7$) which are not naturally reductive. This is achieved by imposing certain symmetry assumptions in the set of all left-invariant metrics on $SO(n)$ and by computing the Ricci tensor for such metrics. The Einstein metrics are obtained as solutions of systems polynomial equations, which we manipulate by symbolic computations using Gr\"obner bases.