On pseudo-Riemannian Ricci-parallel Lie groups which are not Einstein

In this paper, we mainly study left invariant pseudo-Riemannian Ricci-parallel metrics on connected Lie groups which are not Einstein. Following a result of Boubel and B\'erard Bergery, there are two typical types of such metrics, which are characterized by the minimal polynomial of the Ricci operator. Namely, its form is either (X-) (X-) (type I), where C R, or X^2 (type II). Firstly, we obtain a complete description of Ricci-parallel metrics of type I. In particular, such a Ricci-parallel metric is uniquely determined by an Einstein metric and an invariant symmetric parallel complex structure up to isometry and scaling. Then we study Ricci-parallel metric Lie algebras of type II by using double extension process. Surprisingly, we find that every double extension of a metric Abelian Lie algebra is Ricci-parallel and the converse holds for Lorentz Ricci-parallel metric nilpotent Lie algebras of type II. Moreover, we construct infinitely many new explicit examples of Ricci-parallel metric Lie algebras which are not Einstein.