Closed G$_2$-structures on unimodular Lie algebras with non-trivial center
Anna Fino, Alberto Raffero, Francesca Salvatore
TL;DR
The authors classify seven-dimensional unimodular Lie algebras with nontrivial center that admit closed G$_2$-structures by showing such algebras are central extensions of six-dimensional Lie algebras via a closed 2-form, with the G$_2$-form expressed as $\varphi=\widetilde{\omega}\wedge\theta+\rho$ and $d\rho=-\widetilde{\omega}\wedge\omega_0$; when $\omega_0$ is symplectic, the extension is a contactization. They completely classify these algebras up to isomorphism, finding twelve nilpotent and eleven solvable non-nilpotent examples, among which two solvables ($\mathfrak{s}_{10}$ and $\mathfrak{s}_{11}$) are contactizations of 6D symplectic algebras, and identify which admit lattices to produce compact quotients. The paper further analyzes semi-algebraic (homogeneous) Laplacian solitons on these central extensions, proving that, for contactizations of symplectic unimodular algebras, the soliton is always expanding with $\lambda=|\tau|_\varphi^2$, and providing a center-dimension≥2 classification of algebras admitting semi-algebraic solitons. Overall, the work links closed G$_2$-geometry on 7D Lie algebras to central extensions and contactizations, producing new locally homogeneous examples and sharpening the landscape of solitons in this setting.
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.