Closed ${\text {G}}_{2}$-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.

Appendix

In this “Appendix”, we give some details on the computations we did to prove Propositions 4.7, 4.8, 4.10, 5.1, 5.2. We focus on the case $\mathfrak {g}= \mathfrak {so}(3)\oplus \mathfrak {aff}({\mathbb R})\oplus \mathfrak {aff}({\mathbb R})$, as in the remaining cases one can proceed similarly.

Let $\{e_1,\ldots ,e_7\}$ be the basis of $\mathfrak {g}=\mathfrak {so}(3)\oplus \mathfrak {aff}({\mathbb R})\oplus \mathfrak {aff}({\mathbb R})$ described in the proof of Proposition 4.7. Then, the structure equations of $\mathfrak {g}$ with respect to the dual basis $\{e^1,\ldots ,e^7\}$ are the following:

$$\begin{aligned} \left( -e^{23},e^{13},-e^{12},0,-e^{45},0,-e^{67}\right) . \end{aligned}$$

Let us consider a generic 3-form $\phi = \sum _{1\le i<j<k\le 7} \phi _{ijk}e^{ijk}\in \Lambda ^3(\mathfrak {g}^*)$, where $\phi _{ijk}\in {\mathbb R}$. The condition $d\phi =0$ is equivalent to the following system of linear equations in the variables $\{\phi _{ijk}\}$:

$$\begin{aligned} \left\{ \begin{array}{lc} \phi _{i46}=0,~\phi _{i47}=0,~\phi _{i56}=0,~\phi _{i57}=0, &{}\quad i=1,2,3,\\ \phi _{125}+\phi _{345}=0,~\phi _{127}+\phi _{367}=0,&{} \\ \phi _{135}-\phi _{245}=0,~\phi _{137}-\phi _{267}=0, &{}\\ \phi _{145}+\phi _{235}=0,~\phi _{237}+\phi _{167}=0,&{} \\ \phi _{567}+\phi _{457}=0.&{} \end{array} \right. \end{aligned}$$

Solving this system, we obtain the following expression for the generic closed 3-form $\phi$ on $\mathfrak {g}$

$$\begin{aligned} \phi= & {} \phi _{123}e^{123} +\phi _{124}e^{124} -\phi _{345}e^{125} +\phi _{126}e^{126} -\phi _{367}e^{127} +\phi _{134}e^{134} +\phi _{245}e^{135} \\&+\,\phi _{136}e^{136} +\phi _{267}e^{137} -\phi _{235}e^{145} -\phi _{237}e^{167}+\phi _{234}e^{234} +\phi _{235}e^{235} +\phi _{236}e^{236} \\&+\,\phi _{237}e^{237}+\phi _{245}e^{245} +\phi _{267} e^{267} +\phi _{345}e^{345} + \phi _{367}e^{367}+\phi _{456}e^{456}-\phi _{567}e^{457}\\&+\,\phi _{467}e^{467}+\phi _{567}e^{567}. \end{aligned}$$

Using this, we compute $\iota _{e_i}\phi$, $i=1,\ldots ,7,$ and we observe that

$$\begin{aligned} b_\phi (e_5,e_5)= & {} - \phi _{567}\left( {\phi _{235}}^2+{\phi _{245}}^2+{\phi _{345}}^2 \right) e^{1234567}, \\ b_\phi (e_7,e_7)= & {} \phi _{567} \left( {\phi _{237}}^2+{\phi _{267}}^2+{\phi _{367}}^2\right) e^{1234567}. \end{aligned}$$

This implies that $b_\phi$ cannot be definite.