Coclosed $G_2$-structures inducing nilsolitons

TL;DR

This paper investigates conditions under which coclosed $G_2$-structures exist on certain Lie algebras, classifies specific nilpotent cases, and explores their relation to nilsoliton metrics and contact structures.

Contribution

It provides obstructions to coclosed $G_2$-structures on Lie algebras with non-trivial center and classifies 2-step nilpotent cases with such structures, linking them to nilsoliton metrics.

Findings

Coclosed $G_2$-structures induce almost complex structures on certain quotients.

Classification of 2-step nilpotent Lie algebras with coclosed $G_2$-structures.

Existence of coclosed $G_2$-structures inducing nilsoliton metrics on these algebras.

Abstract

We show obstructions to the existence of a coclosed $G_2$-structure on a Lie algebra $\mathfrak g$ of dimension seven with non-trivial center. In particular, we prove that if there exist a Lie algebra epimorphism from $\mathfrak g$ to a six-dimensional Lie algebra $\mathfrak h$, with kernel contained in the center of $\mathfrak g$, then any coclosed $G_2$-structure on $\mathfrak g$ induces a closed and stable three form on $\mathfrak h$ that defines an almost complex structure on $\mathfrak h$. As a consequence, we obtain a classification of the 2-step nilpotent Lie algebras which carry coclosed $G_2$-structures. We also prove that each one of these Lie algebras has a coclosed $G_2$-structure inducing a nilsoliton metric, but this is not true for 3-step nilpotent Lie algebras with coclosed $G_2$-structures. The existence of contact metric structures is also studied.