On $Z$-decomposition of some euclidian lie algebras

TL;DR

This paper employs the Z-decomposition to identify locally symmetric Riemannian metrics on specific low-dimensional Lie groups by analyzing the curvature operator spectrum within restricted classes of Lie algebras.

Contribution

It introduces a method to find locally symmetric metrics on certain Lie groups using Z-decomposition and spectral analysis of the curvature operator.

Findings

Identifies conditions for local symmetry in 4D Lie groups.

Classifies Riemannian Lie groups with specific Lie algebra structures.

Provides explicit analysis for subclasses of 3 and 4-dimensional C-spaces.

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.