Naturally reductive pseudo Riemannian 2-step nilpotent Lie groups

TL;DR

This paper characterizes naturally reductive pseudo-Riemannian 2-step nilpotent Lie groups, exploring cases with nondegenerate and degenerate centers, and constructs examples with detailed geometric and algebraic structures.

Contribution

It provides a complete characterization of such Lie groups with nondegenerate centers and introduces new families of naturally reductive spaces with degenerate centers.

Findings

Complete classification for nondegenerate center cases.

Construction of new naturally reductive spaces with degenerate centers.

Analysis of geometric structures and homogeneous properties.

Abstract

This paper deals with naturally reductive pseudo-Riemannian 2-step nilpotent Lie groups $(N, \la \,,\,\ra_N)$, such that $\la \,,\,\ra_N$ is invariant under a left action. The case of nondegenerate center is completely characterized. In fact, whenever $\la \,,\, \ra_N$ restricts to a metric in the center it is proved here that the simply connected Lie group $N$ can be constructed starting from a real representation $(\pi,\vv)$ of a certain Lie algebra $\ggo$. We study the geometry of $(N, \la \,,\,\ra_N)$ and we find the corresponding naturally reductive homogeneous structure. On the other hand, related to the case of degenerate center we provide another family of naturally reductive spaces, both non compact and also compact examples.