Purely coclosed $G_2$-structures on nilmanifolds

Abstract

We classify 7-dimensional nilpotent Lie groups, decomposable or of 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 [16], 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 ∧ 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.