Purely coclosed $G_2$-structures on nilmanifolds

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 $φ$ which is a pure coclosed $G_2$-structure (that is, it satisfies $d*φ=0$, $φ\wedge dφ=0$) 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.