Nilmanifolds with a calibrated G_2-structure

TL;DR

This paper investigates conditions under which seven-dimensional Lie algebras, especially nilpotent ones, admit calibrated G_2-structures, providing obstructions and a classification of such algebras.

Contribution

It introduces new obstructions to the existence of calibrated G_2-structures and classifies nilpotent Lie algebras that support these structures.

Findings

Obstructions relate to symplectic forms on six-dimensional quotients.

Classification of nilpotent Lie algebras with calibrated G_2-structures.

Conditions involving Lie algebra epimorphisms and centers.

Abstract

We introduce obstructions to the existence of a calibrated G_2-structure on a Lie algebra g of dimension seven, not necessarily nilpotent. In particular, we prove that if there is a Lie algebra epimorphism from g to a six-dimensional Lie algebra h with kernel contained in the center of g, then h has a symplectic form. As a consequence, we obtain a classification of the nilpotent Lie algebras that admit a calibrated G_2-structure.