Closed G$_2$-structures on unimodular Lie algebras with non-trivial center

TL;DR

This paper classifies seven-dimensional unimodular Lie algebras with non-trivial centers that admit closed G$_2$-structures, revealing their structure, connection to contactizations of symplectic Lie algebras, and properties of semi-algebraic solitons.

Contribution

It provides a complete classification of such Lie algebras and characterizes semi-algebraic solitons on their contactizations, advancing understanding of G$_2$-geometry on Lie algebras.

Findings

Six Lie algebras are contactizations of symplectic Lie algebras.

Semi-algebraic solitons on these contactizations are necessarily expanding.

Classification of unimodular Lie algebras with large centers admitting semi-algebraic solitons.

Abstract

We characterize the structure of a seven-dimensional Lie algebra with non-trivial center endowed with a closed G$_2$-structure. Using this result, we classify all unimodular Lie algebras with non-trivial center admitting closed G$_2$-structures, up to isomorphism, and we show that six of them arise as the contactization of a symplectic Lie algebra. Finally, we prove that every semi-algebraic soliton on the contactization of a symplectic Lie algebra must be expanding, and we determine all unimodular Lie algebras with center of dimension at least two that admit semi-algebraic solitons, up to isomorphism.