Purely coclosed $G_2$-structures on nilmanifolds

TL;DR

This paper classifies 7-dimensional nilpotent Lie groups with left-invariant purely coclosed $G_2$-structures, identifying which admit such structures and providing explicit examples where they exist.

Contribution

It offers a complete classification of nilpotent Lie groups with purely coclosed $G_2$-structures, including explicit examples and impossibility results.

Findings

Explicit examples of purely coclosed $G_2$-structures on certain nilpotent Lie groups.

Proof of non-existence of such structures on others.

Complete classification within the specified class of nilpotent Lie groups.

Abstract

We classify 7-dimensional nilpotent Lie groups, decomposable or of nilpotency step at most 4, endowed with left-invariant purely coclosed $G_2$-structures. This is done by going through the list of all 7-dimensional nilpotent Lie algebras given by Gong, providing an example of a left-invariant 3-form $\varphi$ which is a pure coclosed $G_2$-structure (that is, it satisfies $d*\varphi=0$, $\varphi \wedge d\varphi=0$) for those nilpotent Lie algebras that admit them; and by showing the impossibility of having a purely coclosed $G_2$-structure for the rest of them.