Non naturally reductive Einstein metrics on $\SU(N)$ via generalized flag manifolds

TL;DR

This paper constructs new invariant Einstein metrics on the special unitary group (N) that are not naturally reductive, using generalized flag manifolds and symmetry conditions, and investigates their isometry properties.

Contribution

It introduces a method to find non naturally reductive Einstein metrics on (N) via generalized flag manifolds and analyzes their isometry classes.

Findings

New non naturally reductive Einstein metrics on (N)

Method using generalized flag manifolds and symmetry conditions

Analysis of isometry problem for these metrics

Abstract

We obtain new invariant Einstein metrics on the compact Lie group $\SU(N)$ which are not naturally reductive. This is achieved by using the generalized flag manifold $G/K=\SU(k_1+\cdots +k_p)/\s(\U(k_1)\times\cdots\times\U(k_p))$ and by taking an appropriate choice of orthogonal basis of the center of Lie subalgebra $\frak k$ for $K$, which poses certain symmetry conditions to the $\Ad(K)$-invariant metrics of $\SU(N)$. We also study the isometry problem for the Einstein metrics found.