Exact $\hbox {G}_{2}$-structures on unimodular Lie algebras

Abstract

We consider seven-dimensional unimodular Lie algebras $\mathfrak {g}$ admitting exact G$_2$-structures, focusing our attention on those with vanishing third Betti number $b_3(\mathfrak {g})$. We discuss some examples, both in the case when $b_2(\mathfrak {g})\ne 0$, and in the case when the Lie algebra $\mathfrak {g}$ is (2,3)-trivial, i.e., when both $b_2(\mathfrak {g})$ and $b_3(\mathfrak {g})$ vanish. These examples are solvable, as $b_3(\mathfrak {g})=0$, but they are not strongly unimodular, a necessary condition for the existence of lattices on the simply connected Lie group corresponding to $\mathfrak {g}$. More generally, we prove that any seven-dimensional (2,3)-trivial strongly unimodular Lie algebra does not admit any exact G$_2$-structure. From this, it follows that there are no compact examples of the form $(\Gamma \backslash \mathrm {G},\varphi )$, where $\mathrm {G}$ is a seven-dimensional simply connected Lie group with (2,3)-trivial Lie algebra, $\Gamma \subset \mathrm {G}$ is a co-compact discrete subgroup, and $\varphi$ is an exact $\mathrm {G}_2$-structure on $\Gamma \backslash \mathrm {G}$ induced by a left-invariant one on $\mathrm {G}$.

Appendix A.

In this “Appendix”, we explicitly write the matrices associated to the endomorphisms $A_1, A_2, A_3,$ introduced in the proof of Theorem 2.3, in the case when $\mathfrak {n}=\mathfrak {n}_{\scriptscriptstyle 1}$ and $\mathfrak {n}=\mathfrak {n}_{\scriptscriptstyle 2}$. For each of these Lie algebras, we write an ordered basis of the cohomology group $H^k(\mathfrak {n}^*)$, $k=1,2,3,$ and the matrix associated to $A_k$ with respect to that basis. For the sake of convenience, we first impose the conditions on $D\in {{\,\mathrm{Der}\,}}(\mathfrak {n})$ for which $\mathfrak {g}= \mathfrak {n}\rtimes _D\mathbb {R}$ is strongly unimodular.

1.1 Case $\mathfrak {n}=\mathfrak {n}_{\scriptscriptstyle 1}$

For the Lie algebra $\mathfrak {n}_{\scriptscriptstyle 1}= \left( 0,0,0,0,e^{12},e^{34}\right)$ with generic derivation $D_1$ given by (2.2), we have that $\mathfrak {g}=\mathfrak {n}\rtimes _{D_1}\mathbb {R}$ is strongly unimodular if and only if $a_8=-a_1-a_4-a_5$. In this case, we get

• $H^1(\mathfrak {n}_{\scriptscriptstyle 1}) = \langle [e^1],~[e^2],~[e^3],~[e^4]\rangle$,

  $$\begin{aligned} A_1 = -\begin{pmatrix} a_1 &{} a_3 &{} 0 &{} 0 \\ a_2 &{} a_4 &{} 0 &{} 0 \\ 0 &{} 0 &{} a_5 &{} a_7 \\ 0 &{} 0 &{} a_6 &{} -a_1-a_4-a_5 \end{pmatrix}; \end{aligned}$$

• $H^2(\mathfrak {n}_{\scriptscriptstyle 1}) = \langle [e^{13}],[e^{14}],[e^{15}], [e^{23}],[e^{24}],[e^{25}],[e^{36}],[e^{46}] \rangle$,

  $$\begin{aligned} A_2 = \left( \begin{array}{cccccccccc} -a_1 - a_5 &{} -a_7 &{} -a_{11} &{} -a_3 &{} 0 &{} 0 &{} a_{13} &{} 0 \\ -a_6 &{} a_4 +a_5. &{} -a_{12} &{} 0 &{} -a_3 &{} 0 &{} 0 &{} a_{13} \\ 0 &{} 0 &{} -2 a_1 -a_4 &{} 0 &{} 0 &{} -a_3 &{} 0 &{} 0\\ -a_2 &{} 0 &{} 0 &{} -a_4 -a_5 &{} -a_7 &{} -a_{11} &{} a_{14} &{} 0\\ 0 &{} -a_2 &{} 0 &{} -a_6 &{}a _1 +a_5 &{} -a_{12} &{} 0 &{} a_{14} \\ 0 &{} 0 &{} -a_2 &{} 0 &{} 0 &{} -2 a_4 -a_1 &{} 0 &{} 0\\ 0 &{} 0 &{} 0&{} 0 &{} 0&{} 0 &{} a_1+a_4 -a_5 &{} -a_7 \\ 0 &{} 0 &{} 0 &{} 0&{} 0 &{} 0 &{} -a_6 &{} 2 a_4 +2 a_1 +a_5 \end{array}\right) ; \end{aligned}$$

• $H^3(\mathfrak {n}_{\scriptscriptstyle 1}) = \langle [e^{125}], [e^{135}], [e^{136}], [e^{145}], [e^{146}], [e^{235}], [e^{236}], [e^{245}], [e^{246}], [e^{346}] \rangle$,

  $$\begin{aligned} A_3 =\left( \begin{array}{ccccccccccc} 2 a_4 +2 a_1 &{} 0&{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0\\ 0 &{} 2 a_1 +a_5 +a_4 &{} 0 &{} a_6 &{} 0 &{} a_2 &{} 0 &{} 0 &{} 0 &{} 0\\ 0 &{} 0 &{} a_5 {-}a_4 &{} 0 &{} a_6 &{} 0 &{} a_2 &{} 0 &{} 0 &{} 0\\ 0 &{} a_7 &{} 0 &{} a_1 {-}a_5 &{} 0 &{} 0 &{} 0 &{} a_2 &{} 0 &{} 0\\ 0 &{} 0 &{} a_7 &{} 0 &{} {-}2 a_4 {-}a_5 {-}a_1 &{} 0 &{} 0 &{} 0 &{} a_2 &{} 0\\ 0 &{} a_3 &{} 0 &{} 0 &{} 0 &{} 2 a_4 +a_5 +a_1 &{} 0 &{} a_6 &{} 0 &{} 0\\ 0 &{} 0 &{} a_3 &{} 0 &{} 0 &{} 0 &{} a_5 {-}a_1 &{} 0 &{} a_6 &{} 0\\ 0 &{} 0&{} 0 &{} a_3 &{} 0 &{} a_7 &{} 0 &{} a_4 {-}a_5 &{} 0&{} 0\\ 0 &{} 0 &{} 0 &{} 0 &{} a_3 &{} 0 &{} a_7 &{} 0 &{} {-}2 a_1 {-}a_5 {-}a_4 &{} 0\\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} {-}2 a_4 {-}2 a_1 \end{array}\right) . \end{aligned}$$

1.2 Case $\mathfrak {n}=\mathfrak {n}_{\scriptscriptstyle 2}$

Consider the Lie algebra $\mathfrak {n}_{\scriptscriptstyle 2}=\left( 0,0,0,0,e^{13}-e^{24},e^{14}+e^{23}\right)$ with generic derivation $D_2$ given by (2.3). Then, $\mathfrak {g}=\mathfrak {n}_{\scriptscriptstyle 2}\rtimes _{D_2}\mathbb {R}$ is strongly unimodular if and only if $a_7=-a_1$. In this case, we have

• $H^1(\mathfrak {n}_{\scriptscriptstyle 2}) = \langle [e^1],~[e^2],~[e^3],~[e^4] \rangle$,

  $$\begin{aligned} A_1 = \begin{pmatrix} -a_1 &{} a_2 &{} -a_5 &{} a_6\\ -a_2 &{} -a_1 &{} -a_6 &{} -a_5\\ -a_3 &{} a_4 &{} a_1 &{} a_8\\ -a_4 &{} -a_3 &{} -a_8 &{} a_1 \end{pmatrix}; \end{aligned}$$

• $H^2(\mathfrak {n}_{\scriptscriptstyle 2}) = \langle [e^{12}], [e^{34}], [e^{13}+e^{24}], [e^{14}-e^{23}], [e^{16}+e^{25}], [-e^{15}+e^{26}], [e^{36}+e^{45}], [-e^{35}+e^{46}] \rangle$,

  $$\begin{aligned} A_2 =\left( \begin{array}{cccccccccc} -2 a_1&{} -2 a_6 &{} -2 a_5 &{} a_9-a_{14} &{} a_{10}+a_{13} &{} 0 &{} 0 &{} 0\\ a_4 &{} 0 &{} a_8-a_2 &{} - \frac{1}{2} a_{15} &{} \frac{1}{2} a_{11}- \frac{1}{2} a_{16} &{} -a_6 &{} \frac{1}{2} a_{13}+ \frac{1}{2} a_{10} &{} - \frac{1}{2} a_{9}+ \frac{1}{2} a_{14}\\ -a_3 &{} a_2-a_8 &{} 0 &{} - \frac{1}{2} a_{16} &{} \frac{1}{2} a_{12}+ \frac{1}{2} a_{15} &{} -a[ _5 &{} - \frac{1}{2} a_{14}+ \frac{1}{2} a_9 &{} \frac{1}{2} a_{13}+ \frac{1}{2} a_{10}\\ 0 &{} &{} 0 &{} -a_1 &{} a_8+2 a_2 &{} 0 &{} -a_5 &{} a_6\\ 0 &{} 0 &{} 0 &{} -a_8-2 a_2 &{} -a_1 &{} 0 &{} -a_6 &{} -a_5\\ 0 &{} 2 a_4 &{} -2 a_3 &{} 0 &{} 0 &{} 2 a_1 &{} a_{11} -a_{16} &{} a_{12} +a_{15}\\ 0 &{} 0 &{} 0 &{} -a_3 &{} a_4 &{} 0 &{} a_1 &{} a_2 +2 a_8\\ 0 &{} 0 &{} 0 &{} -a_4 &{} -a_3 &{} 0 &{} -a_2 -2 a_8 &{} a_1 \end{array}\right) ; \end{aligned}$$

• $H^3(\mathfrak {n}_{\scriptscriptstyle 2}) = \langle [e^{125}], [e^{126}], [e^{345}], [e^{346}], [-e^{135}+e^{146}], [-e^{135}+e^{236}], [e^{135}+e^{245}], [e^{136}+e^{246}], [e^{145}-e^{246}], [e^{235}-e^{246}] \rangle$,

  $$\begin{aligned} A_3 = \left( \begin{array}{cccccccccc} -2 a_1 &{} 0&{} 0 &{} -a_4 &{} -a_4&{} 0 &{} -a_3 &{} 0 &{} -a_4 &{} a_2+a_8\\ 0 &{} -2a_1 &{} -a_2-a_8 &{} a_5 &{} -a_6 &{} -2 a_5 &{} -a_6 &{} 2 a_6 &{} -a_5 &{} 0\\ 0 &{} a_2+a_8 &{} 2 a_1&{} a_6 &{} a_5&{} 0 &{} -a_5 &{} 0 &{} -a_6 &{} 0\\ a_6 &{} 0 &{} a_4&{} 0 &{} -a_2 -2 a_8 &{} 0 &{} -a_8 -2 a_2 &{} a_2+a_8&{} 0 &{} 0\\ a_5&{} 0 &{} a_3 &{} a_2+2a_8 &{} 0 &{} -a_2-a_8&{} 0 &{} 0 &{} -a_8-2 a_2 &{} 0\\ 0 &{} -a_3 &{} 0 &{} 0 &{} -a_2&{} 0 &{} -a_8 -2 a_2 &{} a_2-a_8 &{} 0 &{} -a_5\\ -a_5 &{} 0 &{} -a_3&{} a_2 &{} 0 &{} a_2+a_8 &{} 0 &{} 0 &{} -a_8 &{} 0\\ 0 &{} -a_4 &{} 0 &{} a_2 &{} 0 &{} -a_2+a_8 &{} 0 &{} 0 &{} -a_8 -2 a_2 &{} a_6\\ -a_6 &{} 0 &{} a_4 &{} 0 &{} a_2 &{} 0 &{} a_8 &{} a_2+a_8&{} 0 &{} 0\\ -a_2-a_8 &{} 0 &{} 0 &{} a_3 &{} a_3 &{} -2 a_3 &{} a_4&{} -2 a_4 &{} -a_3 &{} -2 a_1 \end{array}\right) . \end{aligned}$$