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

Abstract

We obtain new invariant Einstein metrics on the compact Lie groups SO(n) (n 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 Grbner bases.