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

26 pages, 16 tables, Accompanying SageMath worksheets available at http://agt.cie.uma.es/~vicente.munoz/Sage-worksheets.zip