Closed G$_2$-structures on non-solvable Lie groups

TL;DR

This paper explores the existence and classification of left-invariant closed G$_2$-structures on seven-dimensional non-solvable Lie groups, providing new examples and conditions for their existence.

Contribution

It presents the first known examples of closed G$_2$-structures on non-solvable Lie groups and classifies unimodular Lie algebras with such structures.

Findings

Existence of closed G$_2$-structures only when semisimple part is $rak{sl}(2,b{R})$

Unimodular Lie algebras with non-trivial Levi decomposition admitting these structures are classified

Provides the first examples of closed G$_2$-structures on non-solvable Lie groups

Abstract

We investigate the existence of left-invariant closed G$_2$-structures on seven-dimensional non-solvable Lie groups, providing the first examples of this type. When the Lie algebra has trivial Levi decomposition, we show that such a structure exists only when the semisimple part is isomorphic to $\mathfrak{sl}(2,\mathbb{R})$ and the radical is unimodular and centerless. Moreover, we classify unimodular Lie algebras with non-trivial Levi decomposition admitting closed G$_2$-structures.