Homogeneous Einstein metrics on SU(n)

TL;DR

This paper investigates the existence of additional inequivalent Einstein metrics on the Lie group SU(n), providing explicit constructions for multiple such metrics on both even and odd cases.

Contribution

It introduces two ansatz methods to find new inequivalent Einstein metrics on SU(n), expanding the known set of such metrics beyond bi-invariant ones.

Findings

Constructed (2k+1) inequivalent Einstein metrics on SU(2k)

Constructed 2k inequivalent Einstein metrics on SU(2k+1)

Demonstrated the existence of multiple new Einstein metrics on SU(n)

Abstract

It is known that every compact simple Lie group admits a bi-invariant homogeneous Einstein metric. In this paper we use two ansatz to probe the existence of additional inequivalent Einstein metrics on the Lie group SU (n) for arbitrary n. We provide an explicit construction of (2k+1) inequivalent Einstein metrics on SU (2k) and 2k inequivalent Einstein metrics on SU (2k + 1).