Examples of naturally reductive pseudo-Riemannian Lie groups

TL;DR

This paper presents examples of naturally reductive pseudo-Riemannian Lie groups, including a 2-step nilpotent Lie group with a degenerate center and a non-bi-invariant metric, expanding understanding of such geometric structures.

Contribution

It provides explicit examples of naturally reductive pseudo-Riemannian Lie groups with degenerate centers, which were previously not well-documented.

Findings

Constructed a 2-step nilpotent Lie group with a degenerate center.

Demonstrated the existence of naturally reductive pseudo-Riemannian metrics not arising from bi-invariance.

Expanded the class of known examples of naturally reductive pseudo-Riemannian spaces.

Abstract

We provide examples of naturally reductive pseudo-Riemannian spaces, in particular an example of a naturally reductive pseudo-Riemannian 2-step nilpotent Lie group $(N, < \,,\,>_N)$, such that $< \,,\,>_N$ is invariant under a left action and for which the center is degenerate. The metric does not correspond to a bi-invariant one.