Abstract
We show obstructions to the existence of a coclosed
{\mathrm{G}_{2}}
-structure
on a Lie algebra
{\mathfrak{g}}
of dimension seven with non-trivial center.
In particular, we prove that if there exists a Lie algebra epimorphism
from
{\mathfrak{g}}
to a six-dimensional Lie algebra
{\mathfrak{h}}
, with the kernel contained in the center of
{\mathfrak{g}}
, then any coclosed
{\mathrm{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
{\mathrm{G}_{2}}
-structures.
We also prove that each one of these Lie algebras has a coclosed
{\mathrm{G}_{2}}
-structure inducing a nilsoliton metric,
but this is not true
for 3-step nilpotent Lie algebras with coclosed
{\mathrm{G}_{2}}
-structures. The existence of contact metric structures is also studied.
BRANCHED SHADOWS AND COMPLEX STRUCTURES ON 4-MANIFOLDS