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\'{e}rard 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 $(X-\alpha)(X-\bar{\alpha})$ (type I), where $\alpha\in \mathbb{C}\setminus \mathbb{R}$, 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.