On $Z$-decomposition of some euclidian lie algebras

Abstract

In this paper we use the $Z-$decomposition as a tool to find locally symmetric left invariant Riemannian metrics on some Lie groups. For this purpose, we need to compute the spectrum of the curvature operator. Since the study of this spectrum is very difficult, we impose restriction on the class of Riemannian Lie groups and their dimension. We investigate the $4-$dimensional connected Riemannian Lie groups whose associated Lie algebras are $\mathbb{A}_2\oplus 2\mathbb{A}_1,\,\mathbb{A}_{3,1}\oplus \mathbb{A}_1,\, \mathbb{A}_{3,3}\oplus \mathbb{A}_1$ and the subclasses of $3$ and $4-$dimensional Riemannian Lie groups which are $\mathcal{C}-$spaces.