Homogeneous symplectic half-flat 6-manifolds

We consider 6-manifolds endowed with a symplectic half-flat SU(3)-structure and acted on by a transitive Lie group G of automorphisms. We review a classical result of Wolf and Gray allowing one to show the nonexistence of compact non-flat examples. In the noncompact setting, we classify such manifolds under the assumption that G is semisimple. Moreover, in each case, we describe all invariant symplectic half-flat SU(3)-structures up to isomorphism, showing that the Ricci tensor is always Hermitian with respect to the induced almost complex structure. This property of the Ricci tensor is characterized in the general case.