Homogeneous Einstein metrics on SU(n)SU(n)

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)SU(n) for arbitrary nn. We provide an explicit construction of (2k+1)(2k+1) inequivalent Einstein metrics on SU(2k)SU(2k) and 2k2k inequivalent Einstein metrics on SU(2k+1)SU(2k+1).