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

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.

arXiv comment25 pages, to appear in Transformation Groups

References in corpus (14)

• On Low-Dimensional Solvmanifolds
• The Ricci tensor of SU(3)-manifolds
• Laplacian flow for closed G_2 structures: Shi-type estimates, uniqueness and compactness
• Kaehler reduction of metrics with holonomy G_2
• Laplacian flow of homogeneous G2-structures and its solitons
• Nilmanifolds with a calibrated G_2-structure
• Hypersymplectic 4-manifolds, the $G_2$-Laplacian flow and extension assuming bounded scalar curvature
• Spacelike mean curvature flow
• Laplacian Solitons and Symmetry in G_2-geometry
• Calibrated and parallel structures on almost Abelian Lie algebras
• Closed G${}_2$-structures on non-solvable Lie groups
• $S^1$-invariant Laplacian flow
• Remarks on homogeneous solitons of the G${}_2$-Laplacian flow
• Closed $\text{SL}(3,\mathbb{C})$-structures on nilmanifolds