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.

13 pages, to appear in Revista Matematica Complutense

References in corpus (3)

• On the automorphism group of a closed G${}_2$-structure
• Four-dimensional compact solvmanifolds with and without complex analytic structures
• Closed warped G${}_2$-structures evolving under the Laplacian flow

Cited by in corpus (7)

• On the automorphism group of a closed G${}_2$-structure
• A class of eternal solutions to the G${}_2$-Laplacian flow
• Exact $G_2$-structures on unimodular Lie algebras
• Pseudo-Kähler and hypersymplectic structures on semidirect products
• Exact G${}_2$-structures on compact quotients of Lie groups
• On $G_2$-structures, special metrics and related flows
• Closed G${}_2$-structures on unimodular Lie algebras with non-trivial center