# Moduli spaces of (co)closed $\mathrm{G}_{2}$-structures on nilmanifolds

Giovanni Bazzoni  Dipartimento di Scienza ed Alta Tecnologia, Università degli
Studi dell’Insubria, Via Valleggio 11, 22100, Como, Italy
<EMAIL_ADDRESS>and  Alejandro Gil-García  Fachbereich
Mathematik, Universität Hamburg, Bundesstraße 55, 20146 Hamburg, Germany
<EMAIL_ADDRESS>
###### Abstract.

We compute the dimensions of the moduli spaces of left-invariant closed and
coclosed $\mathrm{G}_{2}$-structures on 7-dimensional nilmanifolds, showing
that they are not related to the third Betti number. We also prove that the
group of automorphisms of a coclosed $\mathrm{G}_{2}$-structure is not
abelian, in general.

###### Key words and phrases:

Closed and coclosed $\mathrm{G}_{2}$-structures, moduli spaces, nilmanifolds

###### 2020 Mathematics Subject Classification:

Primary: 53C10, Secondary: 22E25, 53C38, 58D27

##  1\. Introduction and Preliminaries

A $\mathrm{G}_{2}$-structure on a 7-dimensional manifold is a reduction of the
structure group of the frame bundle from $\operatorname{GL}(7,\mathbb{R})$ to
$\mathrm{G}_{2}\subset\operatorname{SO}(7)$; hence, such a manifold is
oriented and Riemannian. The existence of a $\mathrm{G}_{2}$-structure is
topological, and equivalent to the vanishing of the first and second Stiefel-
Whitney class of the manifold (thus to orientability and spinnability); this
is due to Gray [14]. Since $\mathrm{G}_{2}$ is the stabilizer of a (positive)
3-form in $\mathbb{R}^{7}$, a $\mathrm{G}_{2}$-structure on a 7-manifold $M$
can equivalently be defined as a positive 3-form
$\varphi\in\Omega^{3}_{+}(M)$, one for which it is possible to find, at any
point $p\in M$, local coordinates $\\{x_{1},\ldots,x_{7}\\}$ such that

$\varphi_{p}=\mathrm{d}x^{127}+\mathrm{d}x^{347}+\mathrm{d}x^{567}+\mathrm{d}x^{135}-\mathrm{d}x^{236}-\mathrm{d}x^{146}-\mathrm{d}x^{245}\,,$

where
$\mathrm{d}x^{ijk}=\mathrm{d}x_{i}\wedge\mathrm{d}x_{j}\wedge\mathrm{d}x_{k}$.
When one thinks of a $\mathrm{G}_{2}$-structure in this way, the metric and
the orientation are usually denoted $g_{\varphi}$ and
$\mathrm{vol}_{\varphi}$, since they are determined by the positive 3-form
$\varphi$. In fact, $g_{\varphi}$ and $\mathrm{vol}_{\varphi}$ can be
recovered from $\varphi$ via the formula

$6g_{\varphi}(X,Y)\mathrm{vol}_{\varphi}=\iota_{X}\varphi\wedge\iota_{Y}\varphi\wedge\varphi\,,\quad
X,Y\in{\mathfrak{X}}(M)\,.$

$\mathrm{G}_{2}$-structures are related to manifolds with Riemannian holonomy
contained in $\mathrm{G}_{2}$: by a classical result of Fernández and Gray
[8], the metric $g_{\varphi}$ has holonomy contained in $\mathrm{G}_{2}$ if
and only if $\mathrm{d}\varphi=0=\mathrm{d}*_{\varphi}\varphi$, namely, if
$\varphi$ is both closed and coclosed. Notice that $\mathrm{G}_{2}$ is one of
the exceptional holonomy groups appearing in the celebrated Berger list [3].

Furthermore, the differential and the codifferential of a
$\mathrm{G}_{2}$-form $\varphi$ on a 7-manifold $M$ are governed by the so-
called torsion forms $\tau_{i}\in\Omega^{i}(M)$, $i=0,1,2,3$, according to the
following equations (see [4, Proposition 1]):

$\left\\{\begin{array}[]{ccl}\mathrm{d}\varphi&=&\tau_{0}\psi+3\tau_{1}\wedge\varphi+*_{\varphi}\tau_{3}\\\
\mathrm{d}\psi&=&4\tau_{1}\wedge\psi+\tau_{2}\wedge\varphi\end{array}\right.$

where $\psi=*_{\varphi}\varphi$. Thus, there are 16 classes of
$\mathrm{G}_{2}$-structures, and the holonomy reduction is equivalent to the
vanishing of all torsion forms, so that Riemannian manifolds with holonomy
contained in $\mathrm{G}_{2}$ are also known as torsion-free
$\mathrm{G}_{2}$-manifolds. The pure classes are those for which all torsion
forms vanish, but one. We thus get four pure classes:

$\tau_{0}\neq 0$ | nearly parallel | $\mathrm{d}\varphi=\tau_{0}*_{\varphi}\varphi$
---|---|---
$\tau_{1}\neq 0$ | locally conformally parallel |  $\mathrm{d}\varphi=3\tau_{1}\wedge\varphi$, $\mathrm{d}\psi=4\tau_{1}\wedge\psi$
$\tau_{2}\neq 0$ | closed | $\mathrm{d}\psi=\tau_{2}\wedge\varphi$
$\tau_{3}\neq 0$ | purely coclosed | $\mathrm{d}\varphi=*_{\varphi}\tau_{3}\Leftrightarrow\varphi\wedge\mathrm{d}\varphi=0$

Notice that coclosed $\mathrm{G}_{2}$-structures, those for which
$\mathrm{d}\psi=0$, form a class which contains both nearly parallel and
purely coclosed $\mathrm{G}_{2}$-structures.

In this paper we are mainly interested in left-invariant
$\mathrm{G}_{2}$-structures on nilmanifolds. A nilmanifold is the quotient of
a connected, simply connected, nilpotent Lie group $G$ by a lattice $\Gamma$,
i.e. a discrete, cocompact subgroup. Recall that a Lie group $G$ is nilpotent
if and only if its Lie algebra ${\mathfrak{g}}$ is. Nilpotency of a Lie
algebra ${\mathfrak{g}}$ is defined as follows: the lower central series of a
Lie algebra ${\mathfrak{g}}$ is defined by ${\mathfrak{g}}_{0}={\mathfrak{g}}$
and ${\mathfrak{g}}_{i}=[{\mathfrak{g}}_{i-1},{\mathfrak{g}}]$, for $i\geq 1$.
A Lie algebra ${\mathfrak{g}}$ is nilpotent if there exists $m$ such that
${\mathfrak{g}}_{m}=0$. The smallest such $m$ is the nilpotency step of a
nilpotent Lie algebra.

Let $G$ be a connected, simply connected, nilpotent Lie group. Tensors on
${\mathfrak{g}}$ correspond to left-invariant tensors on $G$, which descend to
tensors on the quotient nilmanifold $\Gamma\backslash G$; with a slight abuse
of notation, we call such tensors on $\Gamma\backslash G$ left-invariant. For
instance, left-invariant forms know a great deal about the topology of a
nilmanifold: Nomizu [21] proved that $H^{*}_{dR}(\Gamma\backslash G)$ is
isomorphic to the Lie algebra cohomology of ${\mathfrak{g}}$. A
$\mathrm{G}_{2}$-structure on a nilmanifold $\Gamma\backslash G$ is called
left-invariant if it is induced by a left-invariant 3-form
$\varphi\in\Omega^{3}(\Gamma\backslash G)$; hence, such a
$\mathrm{G}_{2}$-structure can be described by an element
$\varphi\in\Lambda^{3}{\mathfrak{g}}^{*}$.

Nilmanifolds are parallelizable, so that they admit
$\mathrm{G}_{2}$-structures, and also coclosed ones, by a result of Crowley
and Nordström, [6]; not all of them are left-invariant. It is known that
nilmanifolds (not tori) can not admit torsion-free
$\mathrm{G}_{2}$-structures. As for pure, left-invariant
$\mathrm{G}_{2}$-structures on nilmanifolds, we have the following facts:

  * •

they can not be nearly parallel; this is because the metric of a nearly
parallel $\mathrm{G}_{2}$-structure is Einstein [12, Proposition 3.10]. By a
result of Milnor, nilmanifolds (not tori) do not admit left-invariant Einstein
metrics [20, Theorem 2.4].

  * •

they can not be locally conformally parallel; by [16, Theorem A], compact
manifolds with locally conformally parallel $\mathrm{G}_{2}$-structures are
mapping tori of simply connected nearly Kähler 6-manifolds, hence they have
$b_{1}=1$. However, a nilmanifold has $b_{1}\geq 2$.

We are therefore naturally led to consider left-invariant closed and (purely)
coclosed $\mathrm{G}_{2}$-structures on nilmanifolds. The classification of
nilpotent Lie algebras which admit closed $\mathrm{G}_{2}$-structures was
obtained by Conti and Fernández [5]. In [1] Bagaglini, Fino and Fernández
considered coclosed $\mathrm{G}_{2}$-structures on decomposable and 2-step
nilpotent Lie algebras (a Lie algebra is decomposable if it is a direct sum of
two non-trivial Lie algebras). Purely coclosed $\mathrm{G}_{2}$-structures on
2-step nilpotent Lie algebras have been studied in [7]. Purely coclosed
$\mathrm{G}_{2}$-structures on nilpotent Lie algebras which are either
decomposable or have nilpotency step $\leq 4$ have been tackled in [2].

The main focus of this paper is on moduli spaces of closed and coclosed
$\mathrm{G}_{2}$-structures on nilpotent Lie algebras, and of (left-invariant)
closed and coclosed $\mathrm{G}_{2}$-structures on nilmanifolds (we refer to
Sections 2 and 3 for all the relevant definitions). The starting points of our
analysis are the following:

  * •

by a result of Joyce [17], the moduli space of torsion-free
$\mathrm{G}_{2}$-structures on a compact manifold $M$ is a smooth manifold of
dimension $b_{3}(M)$;

  * •

it is known that the moduli spaces of closed and coclosed
$\mathrm{G}_{2}$-structures on a compact manifold are infinite-dimensional; we
include a proof of these facts in Sections 2 and 3.

In Section 2, we show that the dimension of the moduli space
$\mathcal{M}^{\mathrm{c}}({\mathfrak{g}})$ of closed
$\mathrm{G}_{2}$-structures on a Lie algebra ${\mathfrak{g}}$ which admits
such a structure coincides with the dimension of the moduli space of left-
invariant closed $\mathrm{G}_{2}$-structures on any nilmanifold
$\Gamma\backslash G$,
$\mathcal{M}^{\mathrm{c}}_{\mathrm{inv}}(\Gamma\backslash G)$. Furthermore, we
prove that $\mathcal{M}^{\mathrm{c}}({\mathfrak{g}})$ embeds into
$\mathcal{M}^{\mathrm{c}}(\Gamma\backslash G)$, thus it can be considered as a
finite-dimensional approximation of $\mathcal{M}^{\mathrm{c}}(\Gamma\backslash
G)$. Finally, we compute the dimensions of
$\mathcal{M}^{\mathrm{c}}({\mathfrak{g}})$, and we show that there is no
relation with $b_{3}(\Gamma\backslash G)$.

In Section 3 we follow the same train of thoughts in the coclosed setting; the
results we get are tantamount to those obtained in Section 2. As an
application of our study, we show that the automorphism group of a coclosed
$\mathrm{G}_{2}$-structure on a compact manifold need not be abelian. This
differs from the closed situation; indeed, as proved in [22], the automorphism
group of a closed $\mathrm{G}_{2}$-structure on a compact manifold is abelian.

Appendices A and B contain an overview of the computations, which were
executed using SageMath, [23].

Acknowledgements. We are both very thankful to Vicente Cortés for suggesting
this problem and for useful conversations. We are also indebted to Anna Fino
and Vicente Muñoz for valuable comments on a preliminary version of this
paper. G. B. is partially supported by GNSAGA of INdAM and by grants
PID2020-118452GB-I00 and PID2021-126124NB-I00
(MCIN/AEI/10.13039/501100011033). A. G.-G. is supported by the German Science
Foundation (DFG) under Germany’s Excellence Strategy – EXC 2121 “Quantum
Universe” – 390833306.

##  2\. Moduli spaces of closed $\mathrm{G}_{2}$-structures on nilmanifolds

In this section we consider three moduli spaces of closed
$\mathrm{G}_{2}$-structures:

  * •

closed $\mathrm{G}_{2}$-structures on (nilpotent) Lie algebras,

  * •

left-invariant closed $\mathrm{G}_{2}$-structures on nilmanifolds,

  * •

closed $\mathrm{G}_{2}$-structures on 7-manifolds,

and prove some relations among them.

Given a 7-dimensional Lie algebra ${\mathfrak{g}}$ with a closed
$\mathrm{G}_{2}$-structure, we set

$V^{\mathrm{c}}({\mathfrak{g}})\coloneqq\left\\{\varphi\in\Lambda^{3}{\mathfrak{g}}^{*}\mid\text{$\varphi$
is a $\mathrm{G}_{2}$-structure and }\mathrm{d}\varphi=0\right\\}\,.$

Thus $\varphi\in\Lambda^{3}{\mathfrak{g}}^{*}$ is a closed 3-form and the
formula

$6g_{\varphi}(X,Y)\operatorname{vol}_{\varphi}\coloneqq\iota_{X}\varphi\wedge\iota_{Y}\varphi\wedge\varphi$

defines a scalar product $g_{\varphi}$ and a volume
$\operatorname{vol}_{\varphi}$ on ${\mathfrak{g}}$. Since being positive-
definite is an open condition, $V^{\mathrm{c}}({\mathfrak{g}})$ is an open
subset of $Z^{3}({\mathfrak{g}}^{*})$, and its dimension coincides with that
of $Z^{3}({\mathfrak{g}}^{*})$. $\operatorname{Aut}({\mathfrak{g}})$, the
group of Lie algebra automorphisms of ${\mathfrak{g}}$, is a closed subgroup
of $\operatorname{GL}({\mathfrak{g}})$, and acts naturally on
$V^{\mathrm{c}}({\mathfrak{g}})$; we denote by
$\mathcal{O}(\varphi)\cong\operatorname{Aut}({\mathfrak{g}})/\operatorname{Aut}({\mathfrak{g}})_{\varphi}$
the orbit of the $\operatorname{Aut}({\mathfrak{g}})$-action passing through
$\varphi$; here

$\operatorname{Aut}({\mathfrak{g}})_{\varphi}\coloneqq\operatorname{Aut}({\mathfrak{g}})\cap\\{A\in\operatorname{GL}({\mathfrak{g}})\mid
A^{*}\varphi=\varphi\\}\,,$

which is a compact subgroup.

######  Definition 2.1.

The moduli space of closed $\mathrm{G}_{2}$-structures on ${\mathfrak{g}}$ is

$\mathcal{M}^{\mathrm{c}}({\mathfrak{g}})\coloneqq
V^{\mathrm{c}}({\mathfrak{g}})/\operatorname{Aut}({\mathfrak{g}})=\\{\mathcal{O}(\varphi)\mid\varphi\in
V^{c}({\mathfrak{g}})\\}\,.$

$\mathcal{M}^{\mathrm{c}}({\mathfrak{g}})$ is not smooth in general, rather a
stratified space, since the orbits of different elements of
$V^{\mathrm{c}}({\mathfrak{g}})$ under the action of
$\operatorname{Aut}({\mathfrak{g}})$ may have different dimensions.

######  Definition 2.2.

The dimension of $\mathcal{M}^{\mathrm{c}}({\mathfrak{g}})$ is

(1) $\dim\mathcal{M}^{\mathrm{c}}({\mathfrak{g}})\coloneqq\dim
V^{\mathrm{c}}({\mathfrak{g}})-(\dim\operatorname{Aut}({\mathfrak{g}})-\dim\operatorname{Aut}({\mathfrak{g}})_{\varphi})\,,$

where $\mathcal{O}(\varphi)$ is a principal orbit, i.e. one of maximal
dimension; equivalently, the stabilizer of $\varphi$ has minimal dimension.

If $\Gamma\backslash G$ is a nilmanifold, the moduli space of left-invariant
closed $\mathrm{G}_{2}$-structures on $\Gamma\backslash G$ is

$\mathcal{M}^{\mathrm{c}}_{\mathrm{inv}}(\Gamma\backslash
G)\coloneqq\\{\widetilde{\mathcal{O}}(\varphi_{\mathrm{inv}})\mid\varphi_{\mathrm{inv}}\in
V^{\mathrm{c}}_{\mathrm{inv}}(\Gamma\backslash G)\\}\,,$

where $V^{\mathrm{c}}_{\mathrm{inv}}(\Gamma\backslash G)$ is the space of
left-invariant closed $\mathrm{G}_{2}$-structures on $\Gamma\backslash G$,
thus $V^{\mathrm{c}}_{\mathrm{inv}}(\Gamma\backslash G)\cong
V^{\mathrm{c}}({\mathfrak{g}})$. However, the equivalence relation on
$V^{\mathrm{c}}_{\mathrm{inv}}(\Gamma\backslash G)$ which defines
$\widetilde{\mathcal{O}}(\varphi_{\mathrm{inv}})$ differs from the one
defining $\mathcal{O}(\varphi_{\mathrm{inv}})$ on
$V^{\mathrm{c}}({\mathfrak{g}})$. Indeed, the orbit of
$\varphi_{\mathrm{inv}}\in V^{\mathrm{c}}_{\mathrm{inv}}(\Gamma\backslash G)$
is the double coset space

$\widetilde{\mathcal{O}}(\varphi_{\mathrm{inv}})\cong\operatorname{Aut}({\mathfrak{g}},\Gamma)\backslash\operatorname{Aut}({\mathfrak{g}})\big{/}\operatorname{Aut}({\mathfrak{g}})_{\varphi_{\mathrm{inv}}}\,,$

where $\operatorname{Aut}({\mathfrak{g}},\Gamma)$ denotes the space of
automorphisms that preserve the lattice $\Gamma$ (we have used the
identification
$\operatorname{Aut}({\mathfrak{g}})\cong\operatorname{Aut}(G)$). However
$\operatorname{Aut}({\mathfrak{g}},\Gamma)\subset\operatorname{Aut}({\mathfrak{g}})$
is a discrete subgroup, hence its dimension is zero. This implies that
$\dim\widetilde{\mathcal{O}}(\varphi_{\mathrm{inv}})=\dim\mathcal{O}(\varphi_{\mathrm{inv}})$.
We thus have the following proposition:

######  Proposition 2.3.

On a nilmanifold $\Gamma\backslash G$ equipped with a left-invariant closed
$\mathrm{G}_{2}$-structure we have
$\dim\mathcal{M}^{\mathrm{c}}_{\mathrm{inv}}(\Gamma\backslash
G)=\dim\mathcal{M}^{\mathrm{c}}({\mathfrak{g}})$.

######  Definition 2.4.

Let $M$ be a manifold endowed with closed $\mathrm{G}_{2}$-structures. The
moduli space of closed $\mathrm{G}_{2}$-structures on $M$ is

$\mathcal{M}^{\mathrm{c}}(M)\coloneqq\\{\varphi\in\Omega^{3}_{+}(M),\mathrm{d}\varphi=0\\}/\operatorname{Diff}_{0}(M)=\\{\mathcal{O}(\varphi)\mid\varphi\in\Omega^{3}_{+}(M),\mathrm{d}\varphi=0\\}\,,$

where $\mathcal{O}(\varphi)$ is the orbit of $\varphi$ under the action of
$\operatorname{Diff}_{0}(M)$, so that
$\mathcal{O}(\varphi)\cong\operatorname{Diff}_{0}(M)/\operatorname{Diff}_{0}(M)_{\varphi}$.

Here $\operatorname{Diff}_{0}(M)_{\varphi}$ is the stabilizer of $\varphi$ in
$\operatorname{Diff}_{0}(M)$. The following result is well-known, but we
include a proof for completeness. We are grateful to Lothar Schiemanowski for
pointing this argument out to us.

######  Proposition 2.5.

Let $M$ be a compact manifold endowed with closed $\mathrm{G}_{2}$-structures.
Then $\mathcal{M}^{\mathrm{c}}(M)$ is infinite-dimensional.

###### Proof.

$\\{\varphi\in\Omega^{3}_{+}(M),\mathrm{d}\varphi=0\\}$ is an open set in the
infinite-dimensional vector space $Z^{3}(M)$ of closed 3-forms on $M$, acted
on by $\operatorname{Diff}_{0}(M)$. Pick
$[\varphi]\in\mathcal{M}^{\mathrm{c}}(M)$. Then

$T_{[\varphi]}\mathcal{M}^{\mathrm{c}}(M)=Z^{3}(M)/T_{\varphi}(\operatorname{Diff}_{0}(M)\cdot\varphi)\,,$

where
$T_{\varphi}(\operatorname{Diff}_{0}(M)\cdot\varphi)=\\{\mathcal{L}_{X}\varphi\mid
X\in{\mathfrak{X}}(M)\\}$. Since $\mathrm{d}\varphi=0$,

$T_{\varphi}(\operatorname{Diff}_{0}(M)\cdot\varphi)=\\{\mathrm{d}(\iota_{X}\varphi)\mid
X\in{\mathfrak{X}}(M)\\}=\mathrm{d}\left(\Omega^{2}_{7}(M)\right)\,,$

where $\Omega^{2}(M)=\Omega^{2}_{7}(M)\oplus\Omega^{2}_{14}(M)$ under the
natural action of $\mathrm{G}_{2}$. By harmonic theory,
$Z^{3}(M)=\mathcal{H}^{3}(M)\oplus\mathrm{d}\left(\Omega^{2}(M)\right)$, hence
$T_{[\varphi]}\mathcal{M}^{\mathrm{c}}(M)\cong\mathcal{H}^{3}(M)\oplus\mathrm{d}\left(\Omega^{2}_{14}(M)\right)$,
which is infinite-dimensional. ∎

######  Remark 2.6.

The group $\operatorname{Diff}_{0}(M)_{\varphi}$ is non-trivial, in general.
As proved in [22, Theorem 2.1], it is an abelian Lie group with
$\dim\operatorname{Diff}_{0}(M)_{\varphi}\leq\min\\{b_{2}(M),6\\}$.

Closed $\mathrm{G}_{2}$-structures are sometimes considered the symplectic
analogue of $\mathrm{G}_{2}$-structures. Indeed, they are both defined by
closed differential forms satisfying the same non-degeneracy condition, namely
that at each point of the manifold their orbit under the action of the general
linear group in the corresponding tangent space is open; in other words, they
are stable forms in the sense of Hitchin [15]. If $M$ is a compact symplectic
manifold, let us denote by $\mathcal{S}(M)$ the space of symplectic structures
on $M$; it is an open subset of $Z^{2}(M)$, acted on by the diffeomorphism
group $\operatorname{Diff}(M)$ of $M$ and by its identity component
$\operatorname{Diff}_{0}(M)$. Setting

$\mathcal{M}^{\mathrm{s}}(M)\coloneqq\mathcal{S}(M)/\operatorname{Diff}_{0}(M)\,,$

one has that $\mathcal{M}^{\mathrm{s}}(M)$ is a smooth manifold of dimension
$b_{2}(M)$ (see for instance [11, Proposition 3.1]). With respect to this,
closed $\mathrm{G}_{2}$-structures and symplectic structures behave quite
differently.

If $M=\Gamma\backslash G$ is a nilmanifold admitting closed
$\mathrm{G}_{2}$-structures, we have a natural inclusion $\imath\colon
V^{\mathrm{c}}({\mathfrak{g}})\hookrightarrow\\{\varphi\in\Omega^{3}_{+}(M)\mid\mathrm{d}\varphi=0\\}$.

######  Theorem 2.7.

$\imath$ induces an inclusion
$\bar{\imath}\colon\mathcal{M}^{\mathrm{c}}({\mathfrak{g}})\hookrightarrow\mathcal{M}^{\mathrm{c}}(M)$.

###### Proof.

Let $[\varphi_{1}],[\varphi_{2}]\in\mathcal{M}^{\mathrm{c}}({\mathfrak{g}})$
be such that $[\varphi_{1}]=[\varphi_{2}]$ in $\mathcal{M}^{\mathrm{c}}(M)$.
This means that there exists $f\in\operatorname{Diff}_{0}(M)$ such that
$f^{*}\varphi_{2}=\varphi_{1}$. Since the projection $\pi\colon G\to M$ is the
universal cover, we can lift $f$ to a $\Gamma$-periodic diffeomorphism
$\tilde{f}\colon G\to G$ such that
$\tilde{f}^{*}\tilde{\varphi}_{2}=\tilde{\varphi}_{1}$; here
$\tilde{\varphi}_{j}$, $j=1,2$, denotes the left-invariant form on $G$ which
projects to $\varphi_{j}$. Notice that $\tilde{\varphi}_{j}$ induces a left-
invariant Riemannian metric $\tilde{g}_{j}$ on $G$, $j=1,2$, and that
$\tilde{f}\colon(G,\tilde{g}_{1})\to(G,\tilde{g}_{2})$ is an isometry.
Composing with a left translation if necessary, we can suppose that
$\tilde{f}$ fixes the identity of $G$. By a theorem of Wilson [24] (see also
[19, Proposition 1.3]), it follows that
$\tilde{f}\in\mathrm{Aut}(G)\cong\operatorname{Aut}({\mathfrak{g}})$. But this
implies that $[\varphi_{1}]=[\varphi_{2}]$ in
$\mathcal{M}^{\mathrm{c}}({\mathfrak{g}})$. ∎

Let $M$ be a $\mathrm{G}_{2}$-manifold, that is, a 7-manifold with torsion-
free $\mathrm{G}_{2}$-structures, and consider the moduli space of torsion-
free $\mathrm{G}_{2}$-structures on $M$,

$\mathcal{M}(M)\coloneqq\mathcal{T}(M)/\operatorname{Diff}_{0}(M)\,,$

where

$\mathcal{T}(M)=\\{\varphi\in\Omega^{3}_{+}(M)\mid\mathrm{d}\varphi=0=\mathrm{d}*_{\varphi}\varphi\\}\,.$

The following theorem summarizes the properties of $\mathcal{M}(M)$ (see for
instance [18, Theorem 7.5]):

######  Theorem 2.8.

Let $M$ be a compact $\mathrm{G}_{2}$-manifold. Then $\mathcal{M}(M)$ is a
smooth manifold of dimension $b_{3}(M)$. In fact, the period map
$\mathcal{M}(M)\to H^{3}(M;\mathbb{R})$, $[\varphi]\mapsto[\varphi]_{dR}$ is a
local diffeomorphism.

If $M=\Gamma\backslash G$ is a nilmanifold, Theorem 2.7 shows that
$\mathcal{M}^{\mathrm{c}}(M)$ contains a full copy of
$\mathcal{M}^{\mathrm{c}}({\mathfrak{g}})$. We can ask whether this “finite-
dimensional approximation” of $\mathcal{M}^{\mathrm{c}}(M)$ has some
relationship with $b_{3}(M)=b_{3}({\mathfrak{g}})$, where the last equality
follows from Nomizu theorem [21]. We address this question next.

According to (1), to compute $\dim\mathcal{M}^{\mathrm{c}}({\mathfrak{g}})$ we
need to compute $\dim V^{\mathrm{c}}({\mathfrak{g}})$,
$\dim\operatorname{Aut}({\mathfrak{g}})$ and
$\dim\operatorname{Aut}({\mathfrak{g}})_{\varphi}$.

  * •

As we remarked above, $\dim V^{\mathrm{c}}({\mathfrak{g}})=\dim
Z^{3}({\mathfrak{g}}^{*})$;

  * •

to compute $\dim\operatorname{Aut}({\mathfrak{g}})$, we simply compute the
dimension of its Lie algebra
$\operatorname{Der}({\mathfrak{g}})=\\{D\in\mathfrak{gl}({\mathfrak{g}})\mid
D[X,Y]=[DX,Y]+[X,DY]\ \forall~{}X,Y\in{\mathfrak{g}}\\}$;

  * •

to compute $\dim\operatorname{Aut}({\mathfrak{g}})_{\varphi}$ it is enough to
calculate the dimension of its Lie algebra
$\operatorname{Der}({\mathfrak{g}})_{\varphi}=\\{D\in\operatorname{Der}({\mathfrak{g}})\mid
D\cdot\varphi=0\\}$, where, for $X,Y,Z\in{\mathfrak{g}}$,

$(D\cdot\varphi)(X,Y,Z)=-\varphi(DX,Y,Z)-\varphi(X,DY,Z)-\varphi(X,Y,DZ)\,.$

To compute the dimension of $\operatorname{Aut}({\mathfrak{g}})_{\varphi}$ we
need to start from a background closed $\mathrm{G}_{2}$-structure on
${\mathfrak{g}}$. In [5], Conti and Fernández classified the 7-dimensional
NLAs admitting a closed $\mathrm{G}_{2}$-structure: they showed that only 12
of the 7-dimensional NLAs admit such a structure; 3 of them are decomposable
and 9 are indecomposable. For each NLA ${\mathfrak{g}}$ with a closed
$\mathrm{G}_{2}$-structure $\varphi$, the authors exhibit an explicit coframe
$\\{\eta^{1},\ldots,\eta^{7}\\}$ of ${\mathfrak{g}}^{*}$ on which $\varphi$
takes the standard form,

$\varphi=\eta^{127}+\eta^{347}+\eta^{567}+\eta^{135}-\eta^{236}-\eta^{146}-\eta^{245}\,.$

With this background closed $\mathrm{G}_{2}$-structure $\varphi$ we can
compute the dimension of the stabilizer
$\operatorname{Aut}({\mathfrak{g}})_{\varphi}$ for each of the pairs
$({\mathfrak{g}},\varphi)$; as we explained before, to do this, we compute the
dimension of its Lie algebra, $\operatorname{Der}({\mathfrak{g}})_{\varphi}$.
The results are contained in the following table (see Appendix A for the
explicit closed $\mathrm{G}_{2}$-form we use and for an overview of the
computations).

Table 1. Nilpotent Lie algebras with closed $\mathrm{G}_{2}$-structures ${\mathfrak{g}}$ |  | $\dim$ | $\dim$ | $\dim$
---|---|---|---|---
| $V^{\mathrm{c}}({\mathfrak{g}})$ | $\operatorname{Aut}({\mathfrak{g}})$ | $\operatorname{Aut}({\mathfrak{g}})_{\varphi}$
${\mathfrak{n}}_{1}$ | $(0,0,0,0,0,0,0)$ | 35 | 49 | 14
${\mathfrak{n}}_{4}$ | $(0,0,0,0,12,13,0)$ | 27 | 27 | 1
${\mathfrak{n}}_{10}$ | $(0,0,0,12,13,23,0)$ | 27 | 25 | 1
$147E1(2)$ | $\begin{array}[]{c}\left(0,0,0,12,23,-13,\right.\\\ \left.2\cdot(-16+25+26-34)\right)\end{array}$ | 20 | 15 | 1
$247A$ | $(0,0,0,12,13,14,15)$ | 23 | 19 | 0
$247L$ | $(0,0,0,12,13,14+23,15)$ | 23 | 17 | 0
$257A$ | $(0,0,12,0,0,13+24,15)$ | 23 | 19 | 0
$257B$ | $(0,0,12,0,0,13,14+25)$ | 23 | 18 | 0
$1357N(1)$ | $(0,0,12,0,13+24,14,15+23+34+46)$ | 20 | 13 | 0
$1357S(-3)$ | $\begin{array}[]{c}\left(0,0,12,0,13,23+24,\right.\\\ \left.15+16+25-3\cdot 26+34\right)\end{array}$ | 19 | 12 | 0
$12457H$ | $(0,0,12,13,23,15+24,16+34)$ | 19 | 12 | 0
$12457I$ | $(0,0,12,13,23,15+24,16+25+34)$ | 19 | 11 | 0

######  Remark 2.9.

Throughout the paper we use the so-called Salamon’s notation for Lie algebras.
For instance, $(13,-23,0)$ denotes the 3-dimensional Lie algebra whose dual
space has a basis $\\{e^{1},e^{2},e^{3}\\}$ with $\mathrm{d}e^{1}=e^{13}$,
$\mathrm{d}e^{2}=-e^{23}$ and $\mathrm{d}e^{3}=0$. Here $\mathrm{d}$ denotes
the Chevalley-Eilenberg differential, which is the dual map to the Lie
bracket.

######  Remark 2.10.

If ${\mathfrak{g}}={\mathfrak{n}}_{1}$, the abelian Lie algebra, the
automorphism group acts transitively on the space of (closed) 3-forms; the
stabilizer of a point is isomorphic to $\mathrm{G}_{2}$, and the moduli space
is just one point.

As we see from Table 1, for most of the pairs $({\mathfrak{g}},\varphi)$ one
has $\dim\operatorname{Aut}({\mathfrak{g}})_{\varphi}=0$, the stabilizer is
minimal, and the orbit $\mathcal{O}(\varphi)$ has maximal dimension. Hence we
can immediately compute the dimension of the corresponding moduli space. This
is not the case, however, for
${\mathfrak{g}}\in\\{{\mathfrak{n}}_{4},{\mathfrak{n}}_{10},147E1(2)\\}$. To
determine whether the dimension of the given orbit is maximal in each case,
the strategy is to slightly perturb the given $\varphi$ to a closed
$\mathrm{G}_{2}$ form $\varphi_{\varepsilon}$, for $\varepsilon$ sufficiently
small, whose stabilizer has dimension 0.

  * •

In the Lie algebra ${\mathfrak{n}}_{4}$, the orbit of the given $\varphi$
seems to be minimal.

  * •

In the Lie algebra ${\mathfrak{n}}_{10}$ the orbit of the given $\varphi$ has
1-dimensional stabilizer. Considering the closed $\mathrm{G}_{2}$-structure
$\varphi_{\varepsilon}=\varphi+\varepsilon(e^{257}+e^{347})$, one can check
that $\dim\operatorname{Aut}({\mathfrak{n}}_{10})_{\varphi_{\varepsilon}}=0$.

  * •

In the Lie algebra ${\mathfrak{g}}=147E1(2)$ the orbit of the given $\varphi$
has 1-dimensional stabilizer. Considering the closed
$\mathrm{G}_{2}$-structure
$\varphi_{\varepsilon}=\varphi+\varepsilon(e^{156}-e^{237}+e^{346})$, one sees
that $\dim\operatorname{Aut}({\mathfrak{g}})_{\varphi_{\varepsilon}}=0$.

Summing up, we can compute the dimension of
$\mathcal{M}^{\mathrm{c}}({\mathfrak{g}})$ for most of the 7-dimensional
nilpotent Lie algebras with a closed $\mathrm{G}_{2}$ structure. We collect
the results in the following table, together with the third Betti number of
each Lie algebra:

Table 2.  ${\mathfrak{g}}$ | $\dim\mathcal{M}^{\mathrm{c}}({\mathfrak{g}})$ | $b_{3}({\mathfrak{g}})$ | ${\mathfrak{g}}$ | $\dim\mathcal{M}^{\mathrm{c}}({\mathfrak{g}})$ | $b_{3}({\mathfrak{g}})$
---|---|---|---|---|---
${\mathfrak{n}}_{1}$ | 0 | 35 | $257A$ | 4 | 14
${\mathfrak{n}}_{4}$ | $\leq 1$ | 21 | $257B$ | 5 | 13
${\mathfrak{n}}_{10}$ | 3 | 20 | $1357N(1)$ | 7 | 8
$147E1(2)$ | 6 | 10 | $1357S(-3)$ | 7 | 8
$247A$ | 4 | 13 | $12457H$ | 7 | 6
$247L$ | 6 | 13 | $12457I$ | 8 | 6

By looking at Table 2 we deduce the following theorem:

######  Theorem 2.11.

On 7-dimensional non-abelian nilpotent Lie algebras with a closed
$\mathrm{G}_{2}$-structure there is no relation between the dimension of
$\mathcal{M}^{\mathrm{c}}({\mathfrak{g}})$ and $b_{3}({\mathfrak{g}})$. More
precisely:

  * •

$\dim\mathcal{M}^{\mathrm{c}}({\mathfrak{g}})<b_{3}({\mathfrak{g}})$ if
${\mathfrak{g}}$ is decomposable;

  * •

both $\dim\mathcal{M}^{\mathrm{c}}({\mathfrak{g}})<b_{3}({\mathfrak{g}})$ and
$\dim\mathcal{M}^{\mathrm{c}}({\mathfrak{g}})>b_{3}({\mathfrak{g}})$ are
possible on indecomposable Lie algebras.

######  Corollary 2.12.

On a 7-dimensional nilmanifold $M=\Gamma\backslash G$ endowed with left-
invariant closed $\mathrm{G}_{2}$-structures there is no relation between the
dimension of $\mathcal{M}^{\mathrm{c}}_{\mathrm{inv}}(M)$ and $b_{3}(M)$.

######  Remark 2.13.

For nilpotent Lie algebras equipped with closed $\mathrm{G}_{2}$-structures,
there is a relation between the dimension of the moduli space
$\mathcal{M}^{\mathrm{c}}({\mathfrak{g}})$ and the nilpotency step of
${\mathfrak{g}}$. Indeed,
$\dim\mathcal{M}^{\mathrm{c}}({\mathfrak{g}})<b_{3}({\mathfrak{g}})$ if
${\mathfrak{g}}$ has nilpotency step $\leq 4$ and
$\dim\mathcal{M}^{\mathrm{c}}({\mathfrak{g}})>b_{3}({\mathfrak{g}})$ for
${\mathfrak{g}}$ 5-step nilpotent.

######  Remark 2.14.

Although there is no classification of closed $\mathrm{G}_{2}$-structures on
7-dimensional solvable Lie algebras, the same kind of computations can carried
out in the solvable setting. As an example, consider the pair
$({\mathfrak{g}},\varphi)$ from [9], where

  * •

${\mathfrak{g}}=(0,0,13-24,14+23,-15+26,-16-25,0)$ is a 2-step solvable Lie
algebra, and

  * •

$\varphi=e^{134}+e^{467}+e^{156}-e^{127}-e^{357}+e^{245}+e^{236}$ is a closed
$\mathrm{G}_{2}$-structure on ${\mathfrak{g}}$.

In this case we have $\dim V^{\mathrm{c}}({\mathfrak{g}})=19$,
$\dim\operatorname{Aut}({\mathfrak{g}})=11$ and
$\dim\operatorname{Aut}({\mathfrak{g}})_{\varphi}=1$. If we consider the
perturbation $\varphi_{\varepsilon}=\varphi+\varepsilon e^{123}$ we have
$\dim\operatorname{Aut}({\mathfrak{g}})_{\varphi_{\varepsilon}}=0$. Hence
$\dim\mathcal{M}^{\mathrm{c}}({\mathfrak{g}})=8>7=b_{3}({\mathfrak{g}})$.

##  3\. Moduli spaces of coclosed $\mathrm{G}_{2}$-structures on NLAs

In this section we focus on moduli spaces of coclosed
$\mathrm{G}_{2}$-structures, using the same approach as in Section 2. We
consider three moduli spaces of coclosed $\mathrm{G}_{2}$-structures:

  * •

coclosed $\mathrm{G}_{2}$-structures on (nilpotent) Lie algebras,

  * •

left-invariant coclosed $\mathrm{G}_{2}$-structures on nilmanifolds,

  * •

coclosed $\mathrm{G}_{2}$-structures on 7-manifolds.

Given a 7-dimensional NLA ${\mathfrak{g}}$ with a coclosed
$\mathrm{G}_{2}$-structure, we set

$V^{\mathrm{cc}}({\mathfrak{g}})\coloneqq\left\\{\psi\in\Lambda^{4}{\mathfrak{g}}^{*}\mid\text{$\psi$
determines a $\mathrm{G}_{2}$-structure and }\mathrm{d}\psi=0\right\\}\,.$

Observe that starting with a 3-form $\varphi$ defining a
$\mathrm{G}_{2}$-structure, one has $\psi=\ast_{\varphi}\varphi$. The
stabilizer of such 4-form is $\pm\mathrm{G}_{2}$; therefore, the 4-form $\psi$
alone does determine the $\mathrm{G}_{2}$-metric, but not the orientation. An
explicit formula, which gives the metric directly from $\psi$, can be found in
[10, Page 5]. Notice that $V^{\mathrm{cc}}({\mathfrak{g}})$ is an open subset
of $Z^{4}({\mathfrak{g}}^{*})$, hence its dimension coincides with that of
$Z^{4}({\mathfrak{g}}^{*})$. $\operatorname{Aut}({\mathfrak{g}})$ acts
naturally on $V^{\mathrm{cc}}({\mathfrak{g}})$. We denote by
$\mathcal{O}(\psi)\cong\operatorname{Aut}({\mathfrak{g}})/\operatorname{Aut}({\mathfrak{g}})_{\psi}$
the orbit of $\psi\in V^{\mathrm{cc}}({\mathfrak{g}})$ under the
$\operatorname{Aut}({\mathfrak{g}})$-action; here
$\operatorname{Aut}({\mathfrak{g}})_{\psi}$ is the stabilizer of $\psi$;
notice that $\operatorname{Aut}({\mathfrak{g}})_{\psi}$ is compact.

######  Definition 3.1.

The moduli space of coclosed $\mathrm{G}_{2}$-structures on ${\mathfrak{g}}$
is

$\mathcal{M}^{\mathrm{cc}}({\mathfrak{g}})\coloneqq
V^{\mathrm{cc}}({\mathfrak{g}})/\operatorname{Aut}({\mathfrak{g}})=\\{\mathcal{O}(\psi)\mid\psi\in
V^{\mathrm{cc}}({\mathfrak{g}})\\}\,.$

As in the closed case, it is a stratified space, since the orbits may have
different dimensions.

######  Definition 3.2.

The dimension of $\mathcal{M}^{\mathrm{cc}}({\mathfrak{g}})$ is

(2) $\dim\mathcal{M}^{\mathrm{cc}}({\mathfrak{g}})\coloneqq\dim
V^{\mathrm{cc}}({\mathfrak{g}})-(\dim\operatorname{Aut}({\mathfrak{g}})-\dim\operatorname{Aut}({\mathfrak{g}})_{\psi})\,,$

where $\mathcal{O}(\psi)$ is a principal orbit; equivalently, $\psi$ has
minimal stabilizer.

If $\Gamma\backslash G$ is a nilmanifold, the moduli space of left-invariant
coclosed $\mathrm{G}_{2}$-structures on $\Gamma\backslash G$ is

$\mathcal{M}^{\mathrm{cc}}_{\mathrm{inv}}(\Gamma\backslash
G)\coloneqq\\{\widetilde{\mathcal{O}}(\varphi_{\mathrm{inv}})\mid\varphi_{\mathrm{inv}}\in
V^{\mathrm{cc}}_{\mathrm{inv}}(\Gamma\backslash G)\\}\,,$

where $V^{\mathrm{cc}}_{\mathrm{inv}}(\Gamma\backslash G)$ is the space of
left-invariant coclosed $\mathrm{G}_{2}$-structures on $\Gamma\backslash G$,
thus $V^{\mathrm{cc}}_{\mathrm{inv}}(\Gamma\backslash G)\cong
V^{\mathrm{cc}}({\mathfrak{g}})$. Arguing exactly as in Section 2, we see that

######  Proposition 3.3.

On a nilmanifold $\Gamma\backslash G$ equipped with a left-invariant coclosed
$\mathrm{G}_{2}$-structure we have
$\dim\mathcal{M}^{\mathrm{cc}}_{\mathrm{inv}}(\Gamma\backslash
G)=\dim\mathcal{M}^{\mathrm{cc}}({\mathfrak{g}})$.

######  Definition 3.4.

Let $M$ be a manifold endowed with coclosed $\mathrm{G}_{2}$-structures. The
moduli space of coclosed $\mathrm{G}_{2}$-structures on $M$ is

$\mathcal{M}^{\mathrm{cc}}(M)\coloneqq\\{\psi\in\Omega^{4}_{+}(M),\mathrm{d}\psi=0\\}/\operatorname{Diff}_{0}(M)=\\{\mathcal{O}(\psi)\mid\psi\in\Omega^{4}_{+}(M),\mathrm{d}\psi=0\\}\,,$

where $\mathcal{O}(\psi)$ is the orbit of $\psi$ under the action of
$\operatorname{Diff}_{0}(M)$, so that
$\mathcal{O}(\psi)\cong\operatorname{Diff}_{0}(M)/\operatorname{Diff}_{0}(M)_{\psi}$.

We also have:

######  Proposition 3.5.

Let $M$ be a compact manifold endowed with coclosed
$\mathrm{G}_{2}$-structures. Then $\mathcal{M}^{\mathrm{cc}}(M)$ is infinite-
dimensional.

###### Proof.

$\\{\psi\in\Omega^{4}_{+}(M),\mathrm{d}\psi=0\\}$ is an open set in the
infinite-dimensional vector space $Z^{4}(M)$ of closed 4-forms on $M$, acted
on by $\operatorname{Diff}_{0}(M)$. Pick
$[\psi]\in\mathcal{M}^{\mathrm{cc}}(M)$. Then

$T_{[\psi]}\mathcal{M}^{\mathrm{cc}}(M)=Z^{4}(M)/T_{\psi}(\operatorname{Diff}_{0}(M)\cdot\psi)\,,$

where
$T_{\psi}(\operatorname{Diff}_{0}(M)\cdot\psi)=\\{\mathcal{L}_{X}\psi\mid
X\in{\mathfrak{X}}(M)\\}$. Since $\mathrm{d}\psi=0$,

$T_{\psi}(\operatorname{Diff}_{0}(M)\cdot\psi)=\\{\mathrm{d}(\iota_{X}\psi)\mid
X\in{\mathfrak{X}}(M)\\}=\mathrm{d}\left(\Omega^{3}_{7}(M)\right)\,,$

where
$\Omega^{3}(M)=\langle\varphi\rangle\oplus\Omega^{3}_{7}(M)\oplus\Omega^{3}_{27}(M)$
under the natural action of $\mathrm{G}_{2}$. By harmonic theory,
$Z^{4}(M)=\mathcal{H}^{4}(M)\oplus\mathrm{d}\left(\Omega^{3}(M)\right)$, hence
$T_{[\psi]}\mathcal{M}^{\mathrm{cc}}(M)\cong\mathcal{H}^{4}(M)\oplus\mathrm{d}\left(\langle\varphi\rangle\oplus\Omega^{3}_{27}(M)\right)$.
∎

######  Remark 3.6.

Not much seems to be known about the group
$\operatorname{Diff}_{0}(M)_{\psi}$, except that it is non-trivial, in
general. Below, we will show that it needs not be abelian.

If $M=\Gamma\backslash G$ is a nilmanifold admitting left-invariant coclosed
$\mathrm{G}_{2}$-structures, we have a natural inclusion $\imath\colon
V^{\mathrm{cc}}({\mathfrak{g}})\hookrightarrow\\{\psi\in\Omega^{4}_{+}(M)\mid\mathrm{d}\psi=0\\}$.
Arguing as in Theorem 2.7, we obtain:

######  Theorem 3.7.

$\imath$ induces an inclusion
$\bar{\imath}\colon\mathcal{M}^{\mathrm{cc}}({\mathfrak{g}})\hookrightarrow\mathcal{M}^{\mathrm{cc}}(M)$.

As we did in the previous section, we can ask whether, on a nilmanifold
$\Gamma\backslash G$ admitting coclosed $\mathrm{G}_{2}$-structures, there is
any relationship between
$\dim\mathcal{M}^{\mathrm{cc}}({\mathfrak{g}})=\dim\mathcal{M}^{\mathrm{cc}}_{\mathrm{inv}}(\Gamma\backslash
G)$ and $b_{4}(\Gamma\backslash G)=b_{3}(\Gamma\backslash
G)=b_{3}({\mathfrak{g}})$. We address this question next.

By (2) to compute $\dim\mathcal{M}^{\mathrm{cc}}({\mathfrak{g}})$ we need the
dimensions of $V^{\mathrm{cc}}({\mathfrak{g}})$,
$\operatorname{Aut}({\mathfrak{g}})$ and
$\operatorname{Aut}({\mathfrak{g}})_{\psi}$.

  * •

We noticed above that $\dim V^{\mathrm{cc}}({\mathfrak{g}})=\dim
Z^{4}({\mathfrak{g}}^{*})$;

  * •

$\dim\operatorname{Aut}({\mathfrak{g}})$ is computed by the dimension of it
Lie algebra
$\operatorname{Der}({\mathfrak{g}})=\\{D\in\mathfrak{gl}({\mathfrak{g}})\mid
D[X,Y]=[DX,Y]+[X,DY]\ \forall~{}X,Y\in{\mathfrak{g}}\\}$;

  * •

$\dim\operatorname{Aut}({\mathfrak{g}})_{\varphi}$ is computed by the
dimension of its Lie algebra
$\operatorname{Der}({\mathfrak{g}})_{\psi}=\\{D\in\operatorname{Der}({\mathfrak{g}})\mid
D\cdot\psi=0\\}$, where, for $X,Y,Z,W\in{\mathfrak{g}}$,

$\displaystyle(D\cdot\psi)(X,Y,Z,W)=$
$\displaystyle-\psi(DX,Y,Z,W)-\psi(X,DY,Z,W)$
$\displaystyle-\psi(X,Y,DZ,W)-\psi(X,Y,Z,DW)\,.$

To compute the dimension of $\operatorname{Aut}({\mathfrak{g}})_{\psi}$ we
need a background coclosed $\mathrm{G}_{2}$-structure on each of these Lie
algebras.

In [1], Bagaglini, Fino and Fernández classified the 7-dimensional
decomposable and 2-step indecomposable nilpotent Lie algebras admitting a
coclosed $\mathrm{G}_{2}$-structure. According to [13], there are 35
decomposable and 9 indecomposable 2-step 7-dimensional NLAs and, as shown in
[1], only 24 of the decomposable and 7 of the 2-step admit a coclosed
$\mathrm{G}_{2}$-structures.

For an explicit coclosed $\mathrm{G}_{2}$-structure we look at [2], where the
authors construct a (purely) coclosed $\mathrm{G}_{2}$-structure on a
7-dimensional Lie algebra ${\mathfrak{g}}$ starting from a particular kind of
$\mathrm{SU}(3)$-structure $(\omega,\psi_{-})$ on a codimension 1 subspace and
a covector $\eta\in{\mathfrak{g}}^{*}$ which pairs non-trivially with the
center of ${\mathfrak{g}}$, see [2, Theorem 4.1]. With these ingredients, the
$\mathrm{G}_{2}$ 4-form on ${\mathfrak{g}}$ is

$\psi=\frac{1}{2}\omega^{2}+\psi_{-}\wedge\eta\,.$

######  Remark 3.8.

For the Lie algebra ${\mathfrak{n}}_{2}$ we used the coclosed
$\mathrm{G}_{2}$-form $\psi$ constructed in [1]. The reason for this is that
the ones constructed in [2] are purely coclosed $\mathrm{G}_{2}$-structures,
namely coclosed $\mathrm{G}_{2}$-structures for which
$\varphi\wedge\mathrm{d}\varphi=0$ (here $\psi=*_{\varphi}\varphi$). However,
combining [1, Theorem 4.3, Theorem 5.1] and [7, Theorem 1.4] one sees that
${\mathfrak{n}}_{2}$ is the only 7-dimensional 2-step nilpotent Lie algebra
which admits coclosed $\mathrm{G}_{2}$-structures but no purely coclosed
$\mathrm{G}_{2}$-structures.

With this background coclosed $\mathrm{G}_{2}$-structure $\psi$ we can compute
the dimension of the stabilizer $\operatorname{Aut}({\mathfrak{g}})_{\psi}$,
for each of the pairs $({\mathfrak{g}},\psi)$. The results are contained in
the following tables (see Appendix B for the explicit coclosed
$\mathrm{G}_{2}$-form we use and for an overview of the computations).

Table 3. Decomposable nilpotent Lie algebras with coclosed $\mathrm{G}_{2}$-structures - I ${\mathfrak{g}}$ | nilpotency | structure equations | $\dim$ | $\dim$ | $\dim$
---|---|---|---|---|---
step | $V^{\mathrm{cc}}({\mathfrak{g}})$ | $\operatorname{Aut}({\mathfrak{g}})$ | $\operatorname{Aut}({\mathfrak{g}})_{\psi}$
${\mathfrak{n}}_{1}$ | 1 | $(0,0,0,0,0,0,0)$ | 35 | 49 | 14
${\mathfrak{n}}_{2}$ | 2 | $(0,0,0,0,0,12,0)$ | 31 | 34 | 4
${\mathfrak{n}}_{3}$ | 2 | $(0,0,0,0,0,12+34,0)$ | 29 | 29 | 2
${\mathfrak{n}}_{4}$ | 2 | $(0,0,0,0,12,13,0)$ | 29 | 27 | 0
${\mathfrak{n}}_{5}$ | 2 | $(0,0,0,0,12,34,0)$ | 28 | 23 | 1
${\mathfrak{n}}_{6}$ | 2 | $(0,0,0,0,13-24,14+23,0)$ | 28 | 23 | 0
${\mathfrak{n}}_{7}$ | 2 | $(0,0,0,0,12,14+23,0)$ | 28 | 24 | 0
${\mathfrak{n}}_{8}$ | 3 | $(0,0,0,0,12,14+25,0)$ | 27 | 22 | 0
${\mathfrak{n}}_{9}$ | 3 | $(0,0,0,0,12,15+34,0)$ | 26 | 20 | 0
${\mathfrak{n}}_{10}$ | 2 | $(0,0,0,12,13,23,0)$ | 28 | 25 | 0
${\mathfrak{n}}_{11}$ | 3 | $(0,0,0,12,13,24,0)$ | 26 | 19 | 0
${\mathfrak{n}}_{12}$ | 3 | $(0,0,0,12,13,14+23,0)$ | 26 | 20 | 0
${\mathfrak{n}}_{13}$ | 3 | $(0,0,0,12,23,14+35,0)$ | 25 | 16 | 0
${\mathfrak{n}}_{14}$ | 3 | $(0,0,0,12,23,14-35,0)$ | 25 | 16 | 0
${\mathfrak{n}}_{15}$ | 3 | $(0,0,0,12,13,14+35,0)$ | 25 | 17 | 0
${\mathfrak{n}}_{16}$ | 4 | $(0,0,0,12,14,15,0)$ | 25 | 19 | 0
${\mathfrak{n}}_{17}$ | 4 | $(0,0,0,12,14,15+24,0)$ | 25 | 18 | 0
${\mathfrak{n}}_{18}$ | 4 | $(0,0,0,12,14,15+24+23,0)$ | 25 | 16 | 0
${\mathfrak{n}}_{19}$ | 4 | $(0,0,0,12,14,15+23,0)$ | 25 | 17 | 0
Table 4. Decomposable nilpotent Lie algebras with coclosed $\mathrm{G}_{2}$-structures - II ${\mathfrak{g}}$ | nilpotency | structure equations | $\dim$ | $\dim$ | $\dim$
---|---|---|---|---|---
step | $V^{\mathrm{cc}}({\mathfrak{g}})$ | $\operatorname{Aut}({\mathfrak{g}})$ | $\operatorname{Aut}({\mathfrak{g}})_{\psi}$
${\mathfrak{n}}_{20}$ | 4 | $(0,0,0,12,14-23,15+34,0)$ | 24 | 15 | 0
${\mathfrak{n}}_{21}$ | 5 | $(0,0,12,13,23,14+25,0)$ | 24 | 14 | 0
${\mathfrak{n}}_{22}$ | 5 | $(0,0,12,13,23,14-25,0)$ | 24 | 14 | 0
${\mathfrak{n}}_{23}$ | 5 | $(0,0,12,13,23,14,0)$ | 24 | 16 | 0
${\mathfrak{n}}_{24}$ | 5 | $(0,0,12,13,14+23,15+24,0)$ | 23 | 13 | 0
Table 5. Indecomposable 2-step nilpotent Lie algebras with coclosed $\mathrm{G}_{2}$-structures ${\mathfrak{g}}$ | structure equations | $\dim$ | $\dim$ | $\dim$
---|---|---|---|---
$V^{\mathrm{cc}}({\mathfrak{g}})$ | $\operatorname{Aut}({\mathfrak{g}})$ | $\operatorname{Aut}({\mathfrak{g}})_{\psi}$
$17$ | $(0,0,0,0,0,0,12+34+56)$ | 29 | 28 | 4
$37A$ | $(0,0,0,0,12,23,24)$ | 29 | 25 | 0
$37B$ | $(0,0,0,0,12,23,34)$ | 28 | 20 | 0
$37B_{1}$ | $(0,0,0,0,12-34,13+24,14)$ | 28 | 20 | 0
$37C$ | $(0,0,0,0,12+34,23,24)$ | 28 | 23 | 0
$37D$ | $(0,0,0,0,12+34,13,24)$ | 28 | 19 | 0
$37D_{1}$ | $(0,0,0,0,12-34,13+24,14-23)$ | 28 | 19 | 3

######  Remark 3.9.

As it happened in the closed case, if ${\mathfrak{g}}={\mathfrak{n}}_{1}$, the
abelian Lie algebra, the automorphism group acts transitively on the space of
(closed) 4-forms; the stabilizer of a point is isomorphic to
$\pm\mathrm{G}_{2}$, and the moduli space is just one point.

As we see from Tables 3, 4 and 5, most of the pairs $({\mathfrak{g}},\psi)$
have $\dim\operatorname{Aut}({\mathfrak{g}})_{\psi}=0$; therefore, the
stabilizer is minimal, the orbit $\mathcal{O}(\psi)$ has maximal dimension and
we can directly compute the dimension of the corresponding moduli space. This
is not the case, however, for
${\mathfrak{g}}\in\\{{\mathfrak{n}}_{2},{\mathfrak{n}}_{3},{\mathfrak{n}}_{5},17,37D_{1}\\}$.
As in the closed setting, we try to slightly perturb the given $\psi$ to a
coclosed $\mathrm{G}_{2}$ form $\psi_{\varepsilon}$, for $\varepsilon$
sufficiently small, whose stabilizer has dimension 0.

  * •

In the Lie algebras ${\mathfrak{n}}_{2}$ and ${\mathfrak{n}}_{3}$, the given
$\psi$ seems to have minimal stabilizer.

  * •

In the Lie algebra ${\mathfrak{n}}_{5}$ the orbit of the given $\psi$ has
1-dimensional stabilizer. Considering the coclosed $\mathrm{G}_{2}$ 4-form
$\psi_{\varepsilon}=\psi+\varepsilon(e^{2356}+e^{2457})$, one computes that
$\dim\operatorname{Aut}({\mathfrak{n}}_{5})_{\psi_{\varepsilon}}=0$.

  * •

In the Lie algebra ${\mathfrak{g}}=17$ the orbit of the given $\psi$ has
4-dimensional stabilizer. Considering the coclosed $\mathrm{G}_{2}$ 4-form
$\psi_{\varepsilon}=\psi+\varepsilon e^{1234}$, one computes that
$\dim\operatorname{Aut}({\mathfrak{g}})_{\psi_{\varepsilon}}=2$. This
dimension seems to be minimal.

  * •

In the Lie algebra ${\mathfrak{g}}=37D_{1}$ the orbit of the given $\psi$ has
3-dimensional stabilizer. One checks that the coclosed $\mathrm{G}_{2}$ 4-form
$\psi_{\varepsilon}=\psi+\varepsilon(e^{2367}+e^{3456})$ has
$\dim\operatorname{Aut}({\mathfrak{g}})_{\psi_{\varepsilon}}=0$.

All in all, we can compute the dimension of
$\mathcal{M}^{\mathrm{cc}}({\mathfrak{g}})$ for most of the 7-dimensional
decomposable and 2-step indecomposable nilpotent Lie algebras endowed with a
coclosed $\mathrm{G}_{2}$ structure. We collect the results in the following
tables, together with the third Betti number of each Lie algebra; we remark
that since a nilpotent Lie algebra is unimodular, its Lie algebra cohomology
satisfies Poincaré duality.

Table 6. Moduli spaces of coclosed $\mathrm{G}_{2}$-structures on decomposable Lie algebras ${\mathfrak{g}}$ | $\dim\mathcal{M}^{\mathrm{cc}}({\mathfrak{g}})$ | $b_{3}({\mathfrak{g}})$ | ${\mathfrak{g}}$ | $\dim\mathcal{M}^{\mathrm{cc}}({\mathfrak{g}})$ | $b_{3}({\mathfrak{g}})$ | ${\mathfrak{g}}$ | $\dim\mathcal{M}^{\mathrm{cc}}({\mathfrak{g}})$ | $b_{3}({\mathfrak{g}})$
---|---|---|---|---|---|---|---|---
${\mathfrak{n}}_{1}$ | 0 | 35 | ${\mathfrak{n}}_{9}$ | 6 | 12 | ${\mathfrak{n}}_{17}$ | 7 | 11
${\mathfrak{n}}_{2}$ | $\leq 1$ | 25 | ${\mathfrak{n}}_{10}$ | 3 | 20 | ${\mathfrak{n}}_{18}$ | 9 | 11
${\mathfrak{n}}_{3}$ | $\leq 2$ | 19 | ${\mathfrak{n}}_{11}$ | 7 | 14 | ${\mathfrak{n}}_{19}$ | 8 | 11
${\mathfrak{n}}_{4}$ | 2 | 21 | ${\mathfrak{n}}_{12}$ | 6 | 14 | ${\mathfrak{n}}_{20}$ | 9 | 8
${\mathfrak{n}}_{5}$ | 5 | 18 | ${\mathfrak{n}}_{13}$ | 9 | 11 | ${\mathfrak{n}}_{21}$ | 10 | 10
${\mathfrak{n}}_{6}$ | 4 | 18 | ${\mathfrak{n}}_{14}$ | 9 | 11 | ${\mathfrak{n}}_{22}$ | 10 | 10
${\mathfrak{n}}_{7}$ | 5 | 18 | ${\mathfrak{n}}_{15}$ | 8 | 11 | ${\mathfrak{n}}_{23}$ | 8 | 10
${\mathfrak{n}}_{8}$ | 6 | 15 | ${\mathfrak{n}}_{16}$ | 6 | 11 | ${\mathfrak{n}}_{24}$ | 10 | 7
Table 7. Moduli spaces of coclosed $\mathrm{G}_{2}$-structures on indecomposable Lie algebras ${\mathfrak{g}}$ | $\dim\mathcal{M}^{\mathrm{cc}}({\mathfrak{g}})$ | $b_{3}({\mathfrak{g}})$ | ${\mathfrak{g}}$ | $\dim\mathcal{M}^{\mathrm{cc}}({\mathfrak{g}})$ | $b_{3}({\mathfrak{g}})$
---|---|---|---|---|---
$17$ | $\leq 3$ | 14 | $37C$ | 5 | 17
$37A$ | 4 | 18 | $37D$ | 9 | 14
$37B$ | 8 | 16 | $37D_{1}$ | 9 | 14
$37B_{1}$ | 8 | 16 |  |  |

We can isolate the following consequences of Tables 6 and 7:

######  Theorem 3.10.

On a 7-dimensional non-abelian nilpotent Lie algebra ${\mathfrak{g}}$ with a
coclosed $\mathrm{G}_{2}$-structure there is no relation between the
dimensions of $\mathcal{M}^{\mathrm{cc}}({\mathfrak{g}})$ and
$b_{3}({\mathfrak{g}})=b_{4}({\mathfrak{g}})$. More precisely:

  * •

both $\dim\mathcal{M}^{\mathrm{cc}}({\mathfrak{g}})<b_{3}({\mathfrak{g}})$ and
$\dim\mathcal{M}^{\mathrm{cc}}({\mathfrak{g}})>b_{3}({\mathfrak{g}})$ are
possible on decomposable Lie algebras;

  * •

$\dim\mathcal{M}^{\mathrm{cc}}({\mathfrak{g}})<b_{3}({\mathfrak{g}})$ if
${\mathfrak{g}}$ is 2-step indecomposable.

######  Remark 3.11.

In contrast with the closed case, for Lie algebras with coclosed
$\mathrm{G}_{2}$-structure we do not have a relation between the dimension of
the moduli space and the nilpotency step of ${\mathfrak{g}}$. For example, the
Lie algebras ${\mathfrak{n}}_{20}$ and ${\mathfrak{n}}_{23}$ are both 4-step
nilpotent, but

$\dim\mathcal{M}^{\mathrm{cc}}({\mathfrak{n}}_{20})>b_{3}({\mathfrak{n}}_{20})\quad\text{and}\quad\dim\mathcal{M}^{\mathrm{cc}}({\mathfrak{n}}_{23})<b_{3}({\mathfrak{n}}_{23}).$

One can wonder whether the uniform bound
$\dim\mathcal{M}^{\mathrm{cc}}({\mathfrak{g}})<b_{3}({\mathfrak{g}})$ on
indecomposable Lie algebras holds because we restrict to nilpotency step 2.
This is indeed the case, as the following example shows:

######  Example 3.12.

The Lie algebra ${\mathfrak{g}}=13457F=(0,0,12,13,14,23,15+26)$ is
7-dimensional, 5-step nilpotent, has $\dim
V^{\mathrm{cc}}({\mathfrak{g}})=23$, $b_{3}({\mathfrak{g}})=7$ and
$\dim\operatorname{Aut}({\mathfrak{g}})=11$. It carries the (purely) coclosed
$\mathrm{G}_{2}$-structure $\psi=\frac{1}{2}\omega^{2}+\psi_{-}\wedge\eta$
with

  * •

$\omega=e^{12}-e^{35}+e^{46}$;

  * •

$\psi_{-}=e^{134}+e^{136}+e^{145}+e^{156}-e^{236}-e^{245}$;

  * •

$\eta=e^{7}+e^{6}+e^{5}-e^{4}-2e^{3}$.

One checks that $\dim\operatorname{Aut}({\mathfrak{g}})_{\psi}=0$, hence
$\dim\mathcal{M}^{\mathrm{cc}}({\mathfrak{g}})=23-11=12>7=b_{3}({\mathfrak{g}})$.

######  Corollary 3.13.

On a 7-dimensional nilmanifold $M=\Gamma\backslash G$ endowed with left-
invariant coclosed $\mathrm{G}_{2}$-structures there is no relation between
the dimension of $\mathcal{M}^{\mathrm{cc}}_{\mathrm{inv}}(M)$ and $b_{3}(M)$.

As a by-product of our study of the automorphism group of coclosed
$\mathrm{G}_{2}$-structures on nilpotent Lie algebras we obtain the following
result:

######  Theorem 3.14.

The automorphism group of a coclosed $\mathrm{G}_{2}$-structure need not be
abelian.

###### Proof.

Consider the decomposable Lie algebra ${\mathfrak{n}}_{2}$ with the coclosed
$\mathrm{G}_{2}$-structure $\psi$ given in Appendix B. As we saw in Table 3,
the Lie algebra $\operatorname{Der}({\mathfrak{n}}_{2})_{\psi}$ has dimension
4; moreover, it contains $\mathfrak{sl}_{2}(\mathbb{R})$ as a subalgebra,
hence it is not abelian. Then $\operatorname{Aut}({\mathfrak{n}}_{2})_{\psi}$
is also not abelian. Notice that
${\mathfrak{n}}_{2}\cong\mathbb{R}^{4}\oplus{\mathfrak{h}}_{3}$, where
${\mathfrak{h}}_{3}$ denotes the Heisenberg Lie algebra. One can pick two non-
commuting derivations in $\operatorname{Der}({\mathfrak{n}}_{2})_{\psi}$ to
give the following automorphisms of the abelian factor $\mathbb{R}^{4}$ in
$\mathbb{R}^{4}\times H_{3}$:

$A_{t}=\begin{pmatrix}\cos t&0&\sin t&0\\\ 0&\cos t&0&-\sin t\\\ -\sin
t&0&\cos t&0\\\ 0&\sin t&0&\cos t\end{pmatrix}\quad\mathrm{and}\quad
B_{s}=\begin{pmatrix}\cos s&0&0&-\sin s\\\ 0&\cos s&-\sin s&0\\\ 0&\sin s&\cos
s&0\\\ \sin s&0&0&\cos s\end{pmatrix}.$

Choosing $t=s=\frac{\pi}{2}$, we obtain two non-commuting automorphisms of the
Lie group $\mathbb{R}^{4}$ which preserve the standard lattice
$\mathbb{Z}^{4}\subset\mathbb{R}^{4}$. Together with the identity on the
factor $H_{3}$, which preserves any given lattice $\Gamma\subset H_{3}$, they
define non-commuting elements in $\operatorname{Diff}(M)_{\psi}$, where $M$ is
the nilmanifold $(\mathbb{Z}^{4}\times\Gamma)\backslash(\mathbb{R}^{4}\times
H_{3})$. ∎

##  Appendix A The closed package

Table 8. Closed $\mathrm{G}_{2}$-structures on 7-dimensional nilpotent Lie algebras ${\mathfrak{g}}$ | $\varphi$
---|---
${\mathfrak{n}}_{1}$ | $e^{127}+e^{347}+e^{567}+e^{135}-e^{146}-e^{236}-e^{245}$
${\mathfrak{n}}_{4}$ | $e^{123}+e^{147}+e^{156}+e^{245}+e^{267}-e^{346}+e^{357}$
${\mathfrak{n}}_{10}$ | $e^{123}+e^{145}+e^{167}-e^{246}+2e^{257}+e^{347}+e^{356}$
$1357N(1)$ | $\begin{array}[]{c}12\sqrt{3}\left(e^{126}+e^{134}+e^{135}-e^{136}-e^{157}+e^{237}\right.\\\ \left.-e^{245}-e^{267}-e^{356}+e^{467}\right)\end{array}$
$1357S(-3)$ | $\begin{array}[]{c}-14\sqrt{70}\left(2e^{123}+2e^{146}-e^{157}+e^{167}+2e^{245}+2e^{246}\right.\\\ \left.+e^{257}+3e^{267}+e^{347}+4e^{356}+2e^{456}\right)\end{array}$
$147E1(2)$ | $\begin{array}[]{c}-6\sqrt{6}\left(2e^{123}-2e^{124}+2e^{135}-e^{157}-e^{167}\right.\\\ \left.+2e^{236}+e^{257}+e^{347}+2e^{456}\right)\end{array}$
$257A$ | $e^{124}-e^{135}-e^{167}+e^{236}-e^{257}-e^{347}+e^{456}$
$257B$ | $-e^{125}+e^{134}+e^{167}-e^{237}-e^{246}-e^{356}+e^{457}$
$247A$ | $e^{123}+e^{145}+e^{167}-e^{246}+e^{257}+e^{347}+e^{356}$
$247L$ | $e^{123}+e^{126}-e^{137}-e^{145}-2e^{167}+e^{246}-e^{257}+e^{347}+e^{356}$
$12457H$ | $e^{125}+e^{127}+e^{136}+e^{147}+e^{234}-e^{256}-e^{267}+e^{357}-e^{456}$
$12457I$ | $\begin{array}[]{c}12\sqrt{6}\left(e^{124}-e^{125}-e^{127}-e^{136}-e^{147}-e^{234}\right.\\\ \left.+e^{267}-e^{345}-e^{357}+e^{456}\right)\end{array}$

######  Remark A.1.

With the frame given in [5], the metrics induced by $\varphi$ on the Lie
algebras $1357S(-3)$, $257A$, $257B$ and $12457H$ appear to be negative-
definite. Also, there seems to be a small typo in the construction of the
$\mathrm{G}_{2}$-structure on ${\mathfrak{n}}_{10}$: to obtain a positive-
definite metric, one should take
$\psi_{2}^{+}=e^{123}+e^{145}-e^{246}+e^{356}$ in the proof of Theorem 4 in
[5].

Next we add, for a particular case, the SageMath 9.3 code used in the article.
To compute the Betti numbers, we set the boundary operator of the Lie algebra
and then we compute the homology with the next code.

⬇

############################## Betti numbers of n4

E = ExteriorAlgebra(SR,’e’,7);

E.inject_variables();

alg_nil_g2 = {”\\\frn_4”:{(0,1):e4,(0,2):e5}};

d = E.coboundary(alg_nil_g2[”\\\frn_4”]);

cplx_dR = d.chain_complex();

cplx_dR.homology()

To compute the dimension of $V^{\mathrm{c}}({\mathfrak{g}})$ we compute the
dimension of $Z^{3}({\mathfrak{g}}^{*})$.

⬇

############################## Dimension of V^c(n4)

E = ExteriorAlgebra(SR,’e’,7);

E.inject_variables();

alg_nil_g2 = {”\\\frn_4”:{(0,1):e4,(0,2):e5}};

d = E.coboundary(alg_nil_g2[”\\\frn_4”]);

varphicoeff = var([”varphi%d%d%d”%(i,j,k)\

for i in range(len(E.basis(1)))\

for j in range(i+1, len(E.basis(1)))\

for k in range(j+1, len(E.basis(1)))]);

varphi = sum(varphicoeff[l]*list(E.basis(3))[l] for l in
range(len(E.basis(3))));

eq0 = [d(varphi).interior_product(c).constant_coefficient()==0 for c in
E.basis(4)];

m = matrix([[lhs.coefficient(x) for x in varphicoeff] for e in eq0 for lhs in
[e.lhs()]]);

show(dim(m.right_kernel()));

The following code compute the dimension of the automorphism group.

⬇

############################## Dimension of Aut(n4)

alg_nil_g2 = {”\\\frn_4”:{(’e0’,’e1’):{’e4’:-1},(’e0’,’e2’):{’e5’:-1}}};

L.<e0,e1,e2,e3,e4,e5,e6> = LieAlgebra(SR,alg_nil_g2[”\\\frn_4”]);

show(len(L.derivations_basis()));

Finally, to compute the dimension of the stabilizer, we use the following
code.

⬇

############################## Derivations of n4

alg_nil_g2 = {”\\\frn_4”:{(’e0’,’e1’):{’e4’:-1},(’e0’,’e2’):{’e5’:-1}}};

L.<e0,e1,e2,e3,e4,e5,e6> = LieAlgebra(SR,alg_nil_g2[”\\\frn_4”]);

d = len(L.derivations_basis());

X = [var(”x%d”%i) for i in range(d)];

Dmat = sum(X[i]*L.derivations_basis()[i] for i in range(d));

############################## Closed G2-form of n4

E = ExteriorAlgebra(SR,’e’,7);

E.inject_variables();

alg_nil_g2 = {”\\\frn_4”:{(0,1):e4,(0,2):e5}};

d = E.coboundary(alg_nil_g2[”\\\frn_4”]);

var([”f%d”%(i) for i in range(len(E.basis(1)))]);

omega = e0*e3+e1*e5+e2*e4;

eta = e6;

psi = e0*e1*e2+e0*e4*e5+e1*e3*e4-e2*e3*e5;

varphi = omega*eta+psi;

############################## Dimension stabilizer

D = E.lift_morphism(Dmat);

eq = [];

for i in range(len(E.gens())):

for j in range(i+1,len(E.gens())):

for k in range(j+1,len(E.gens())):

eq1 =
(varphi.interior_product(D(E.gens()[i])).interior_product(E.gens()[j]).interior_product(E.gens()[k])).constant_coefficient();

eq2 =
(varphi.interior_product(E.gens()[i]).interior_product(D(E.gens()[j])).interior_product(E.gens()[k])).constant_coefficient();

eq3 =
(varphi.interior_product(E.gens()[i]).interior_product(E.gens()[j]).interior_product(D(E.gens()[k]))).constant_coefficient();

eq.append(eq1+eq2+eq3);

eq0 = eq;

eq = [eq0[i]==0 for i in range(len(eq))];

m = matrix([[lhs.coefficient(x) for x in X] for e in eq for lhs in
[e.lhs()]]);

show(dim(m.right_kernel()));

##  Appendix B The coclosed package

Table 9. Coclosed $\mathrm{G}_{2}$-structures on decomposable nilpotent Lie algebras ${\mathfrak{g}}$ | $\psi$
---|---
${\mathfrak{n}}_{1}$ | $e^{1234}+e^{1256}+e^{1367}+e^{1457}+e^{2357}-e^{2467}+e^{3456}$
${\mathfrak{n}}_{2}$ | $-e^{1234}+e^{1257}+e^{1356}-e^{1467}+e^{2367}+e^{2456}-e^{3457}$
${\mathfrak{n}}_{3}$ | $-e^{1234}+e^{1256}+e^{1357}+e^{1467}-e^{2367}+e^{2457}-e^{3456}$
${\mathfrak{n}}_{4}$ | $-2e^{1234}-e^{1256}+e^{1357}+3e^{1456}-e^{1467}-2e^{2356}+e^{2367}+e^{2457}+e^{3467}$
${\mathfrak{n}}_{5}$ | $-e^{1234}+e^{1257}+e^{1267}-e^{1356}+e^{1467}-e^{2367}-e^{2456}-e^{3457}$
${\mathfrak{n}}_{6}$ | $e^{1234}+e^{1256}+e^{1257}+e^{1267}+e^{1467}-e^{2367}+e^{3456}-e^{3457}$
${\mathfrak{n}}_{7}$ | $e^{1234}-e^{1257}-e^{1356}-2e^{1467}+e^{2367}-e^{2456}+e^{3457}$
${\mathfrak{n}}_{8}$ | $\begin{array}[]{c}2e^{1234}+e^{1245}-2e^{1345}-3e^{1236}-e^{2346}+e^{1256}-6e^{1356}+e^{2456}\\\ +e^{2347}+e^{1357}-e^{3457}+e^{2367}+e^{1467}-e^{2567}\end{array}$
${\mathfrak{n}}_{9}$ | $e^{1234}-e^{1257}-e^{1356}-2e^{1467}+e^{2367}-e^{2456}+e^{3457}$
${\mathfrak{n}}_{10}$ | $\begin{array}[]{c}\tfrac{6}{5}e^{1235}-\tfrac{34}{25}e^{1237}+e^{1256}+\tfrac{12}{25}e^{1257}+\tfrac{8}{5}e^{1345}+e^{1346}\\\ -\tfrac{12}{25}e^{1347}-\tfrac{16}{25}e^{1457}+2e^{2345}-\tfrac{6}{5}e^{2347}+e^{2467}-e^{3567}\end{array}$
${\mathfrak{n}}_{11}$ | $e^{2345}-e^{1236}+e^{1246}-2e^{1456}-e^{1237}+e^{1357}-e^{1457}-e^{3467}-e^{2567}$
${\mathfrak{n}}_{12}$ | $\begin{array}[]{c}-35e^{1234}+35e^{1245}+7e^{2345}+14e^{1236}-7e^{1346}+7e^{1256}+7e^{2356}-7e^{1456}\\\ +e^{1237}+e^{1347}+e^{3457}+e^{1367}+e^{2467}+e^{1567}\end{array}$
${\mathfrak{n}}_{13}$ | $-2e^{1245}-2e^{1345}-2e^{2345}-\frac{1}{2}e^{1236}-2e^{2456}-e^{1247}+e^{2357}-e^{3467}+e^{1567}$
${\mathfrak{n}}_{14}$ | $-e^{1345}-e^{1236}-e^{2456}-e^{1247}+e^{2357}-e^{3467}+e^{1567}$
${\mathfrak{n}}_{15}$ | $\begin{array}[]{c}e^{1234}+e^{1345}+\frac{1}{2}e^{2345}-\frac{1}{2}e^{1346}-e^{2346}+\frac{3}{4}e^{1256}\\\\[7.11317pt] -e^{1237}+e^{2457}+e^{2367}+e^{1467}-e^{3567}\end{array}$
${\mathfrak{n}}_{16}$ | $3e^{1235}+2e^{1245}-e^{2345}-e^{1346}-e^{1256}-e^{2456}-e^{1247}+e^{3457}+e^{2367}+e^{1567}$
${\mathfrak{n}}_{17}$ | $-2e^{1235}-e^{1245}+e^{1346}-e^{2456}-e^{1237}+e^{3457}+e^{2367}+e^{2467}-e^{1567}$
${\mathfrak{n}}_{18}$ | $\begin{array}[]{c}-\frac{1}{2}e^{1234}-\frac{1}{2}e^{1245}+\frac{1}{2}e^{1345}+\frac{1}{2}e^{2345}-e^{1236}+\frac{1}{2}e^{1246}-\frac{1}{2}e^{1346}+e^{2346}\\\\[1.5pt] -e^{1356}-\frac{1}{2}e^{2456}+e^{1237}+e^{1257}-e^{2357}-e^{3457}-e^{1267}-e^{2367}+e^{1467}\end{array}$
${\mathfrak{n}}_{19}$ | $\begin{array}[]{c}-\frac{5}{12}e^{1234}-\frac{1}{3}e^{1235}+\frac{1}{2}e^{1245}+\frac{5}{6}e^{1346}-2e^{2346}+\frac{5}{12}e^{1356}-2e^{1456}\\\\[1.5pt] -\frac{3}{5}e^{2456}+e^{1247}+e^{3457}+2e^{2367}-e^{1567}\end{array}$
${\mathfrak{n}}_{20}$ | $-3e^{1234}+\frac{3}{2}e^{1345}-\frac{3}{2}e^{1236}-e^{1356}-\frac{3}{4}e^{2456}-e^{1257}+e^{3457}+e^{2367}-e^{1467}$
${\mathfrak{n}}_{21}$ | $\frac{5}{4}e^{1245}+e^{1346}-\frac{1}{2}e^{2346}-\frac{1}{2}e^{1356}-e^{2356}-e^{1237}-e^{3457}+e^{2467}+e^{1567}-e^{2567}$
${\mathfrak{n}}_{22}$ | $e^{1245}-e^{2346}-e^{1356}-e^{1237}-e^{1347}-e^{2357}-2e^{3457}+e^{1467}-e^{2567}$
${\mathfrak{n}}_{23}$ | $e^{1245}-e^{2346}-e^{1356}+e^{1237}+e^{3457}-e^{1467}+e^{2567}$
${\mathfrak{n}}_{24}$ | $e^{1245}+e^{1346}-e^{2356}+e^{1456}+2e^{1237}-2e^{1257}+2e^{3457}-e^{2467}-e^{1567}$
Table 10. Coclosed $\mathrm{G}_{2}$-structures on indecomposable 2-step nilpotent Lie algebras ${\mathfrak{g}}$ | $\psi$
---|---
$17$ | $-\tfrac{1}{2}e^{1234}+e^{1256}+e^{1357}+e^{1467}-e^{2367}+e^{2457}-\tfrac{1}{2}e^{3456}$
$37A$ | $-e^{1234}+e^{1256}+e^{1356}+e^{1367}-e^{1457}+e^{2357}+e^{2456}+e^{2467}-e^{3456}$
$37B$ | $-e^{1234}-e^{1257}-e^{1367}+e^{1456}-e^{1467}-e^{2356}+e^{2367}-e^{2467}+e^{3457}$
$37B_{1}$ | $2e^{1234}-e^{1267}+2e^{1357}-2e^{1456}-2e^{2356}-2e^{2457}-2e^{3467}$
$37C$ | $-e^{1234}-e^{1267}-e^{1356}+e^{1457}-e^{2357}-e^{2456}+e^{3467}$
$37D$ | $-e^{1234}+e^{1267}-e^{1356}+e^{1357}+e^{1456}-2e^{1457}+e^{2357}-e^{2456}-e^{3467}$
$37D_{1}$ | $e^{1234}+e^{1267}-e^{1357}-e^{1456}-e^{2356}+e^{2456}+e^{3467}$

We include here, again for a particular example, the codes to compute the data
for coclosed $\mathrm{G}_{2}$-structures. The codes to compute the Betti
numbers and the dimension of the automorphism group are the same, but changing
the Lie algebra. To compute the dimension of $V^{\mathrm{cc}}({\mathfrak{g}})$
we use the following code that computes the dimension of
$Z^{4}({\mathfrak{g}}^{*})$.

⬇

############################## Dimension of V^cc(n3)

E = ExteriorAlgebra(SR,’e’,7);

E.inject_variables();

Two_step_7nil = {”\\\frn_3”:{(0,1):e5,(2,3):e5}};

d = E.coboundary(Two_step_7nil[”\\\frn_3”]);

psicoeff = var([”psi%d%d%d%d”%(i,j,k,l)\

for i in range(len(E.basis(1)))\

for j in range(i+1,len(E.basis(1)))\

for k in range(j+1,len(E.basis(1)))\

for l in range(k+1,len(E.basis(1)))]);

psi = sum(psicoeff[m]*list(E.basis(4))[m] for m in range(len(E.basis(4))));

eq0 = [d(psi).interior_product(c).constant_coefficient()==0 for c in
E.basis(5)];

m = matrix([[lhs.coefficient(x) for x in psicoeff] for e in eq0 for lhs in
[e.lhs()]]);

show(dim(m.right_kernel()));

To conclude, the next code compute the dimension of the stabilizer for some
closed 4-form.

⬇

############################## Derivations of n3

Two_step_7nil = {”\\\frn_3”:{(’e0’,’e1’):{’e5’:-1},(’e2’,’e3’):{’e5’:-1}}};

L.<e0,e1,e2,e3,e4,e5,e6> = LieAlgebra(SR,Two_step_7nil[”\\\frn_3”]);

d = len(L.derivations_basis());

X = [var(”x%d”%i) for i in range(d)];

Dmat = sum(X[i]*L.derivations_basis()[i] for i in range(d));

############################## Coclosed G2-form of n3

E = ExteriorAlgebra(SR,’e’,7);

E.inject_variables();

Two_step_7nil = {”\\\frn_3”:{(0,1):e5,(2,3):e5}};

d = E.coboundary(Two_step_7nil[”\\\frn_3”]);

omega = e0*e2+e1*e3+e4*e5;

psi = e0*e1*e4-e0*e3*e5+e1*e2*e5-e2*e3*e4;

eta = e6;

starphi = (1/2)*(omega*omega)+psi*eta;

############################## Dimension stabilizer

D = E.lift_morphism(Dmat);

eq = [];

for i in range(len(E.gens())):

for j in range(i+1,len(E.gens())):

for k in range(j+1,len(E.gens())):

for l in range(k+1,len(E.gens())):

eq1 =
(starphi.interior_product(D(E.gens()[i])).interior_product(E.gens()[j]).interior_product(E.gens()[k]).interior_product(E.gens()[l])).constant_coefficient();

eq2 =
(starphi.interior_product(E.gens()[i]).interior_product(D(E.gens()[j])).interior_product(E.gens()[k]).interior_product(E.gens()[l])).constant_coefficient();

eq3 =
(starphi.interior_product(E.gens()[i]).interior_product(E.gens()[j]).interior_product(D(E.gens()[k])).interior_product(E.gens()[l])).constant_coefficient();

eq4 =
(starphi.interior_product(E.gens()[i]).interior_product(E.gens()[j]).interior_product(E.gens()[k]).interior_product(D(E.gens()[l]))).constant_coefficient();

eq.append(eq1+eq2+eq3+eq4);

eq0 = eq;

eq = [eq0[i]==0 for i in range(len(eq))];

m = matrix([[lhs.coefficient(x) for x in X] for e in eq for lhs in
[e.lhs()]]);

show(dim(m.right_kernel()));

## References

  * [1] L. Bagaglini, M. Fernández, and A. Fino.  Coclosed $\mathrm{G}_{2}$-structures inducing nilsolitons.  Forum Math., 30(1):109–128, 2018.
  * [2] G. Bazzoni, A. Garvín, and V. Muñoz.  Purely coclosed $\mathrm{G}_{2}$-structures on nilmanifolds.  Mathematische Nachrichten, 296(6):2236–2257, 2023.
  * [3] M. Berger.  Sur les groupes d’holonomie homogène des variétés à connexion affine et des variétés riemanniennes.  Bull. Soc. Math. France, 83:279–330, 1955.
  * [4] R. L. Bryant.  Some remarks on $\mathrm{G}_{2}$-structures.  In Proceedings of Gökova Geometry-Topology Conference 2005, pages 75–109. Gökova Geometry/Topology Conference (GGT), Gökova, 2006.
  * [5] D. Conti and M. Fernández.  Nilmanifolds with a calibrated $\mathrm{G}_{2}$-structure.  Differential Geom. Appl., 29(4):493–506, 2011.
  * [6] D. Crowley and J. Nordström.  New invariants of $\mathrm{G}_{2}$-structures.  Geom. Topol., 19(5):2949–2992, 2015.
  * [7] V. del Barco, A. Moroianu, and A. Raffero.  Purely coclosed $\mathrm{G}_{2}$-structures on 2-step nilpotent Lie groups.  Rev. Mat. Complut., 35(2):323–359, 2022.
  * [8] M. Fernández and A. Gray.  Riemannian manifolds with structure group $\mathrm{G}_{2}$.  Ann. Mat. Pura Appl. (4), 132:19–45, 1982.
  * [9] M. Fernández.  A family of compact solvable $\mathrm{G}_{2}$-calibrated manifolds.  Tohoku Math. J. (2), 39(2):287–289, 1987.
  * [10] A. Fino, L. Martín-Merchán, and A. Raffero.  The twisted $\mathrm{G}_{2}$ equation for strong $\mathrm{G}_{2}$-structures with torsion, 2023.
  * [11] J. Fricke and L. Habermann.  On the geometry of moduli spaces of symplectic structures.  Manuscripta Math., 109(4):405–417, 2002.
  * [12] Th. Friedrich, I. Kath, A. Moroianu, and U. Semmelmann.  On nearly parallel $\mathrm{G}_{2}$-structures.  J. Geom. Phys., 23(3-4):259–286, 1997.
  * [13] M.-P. Gong.  Classification of Nilpotent Lie Algebras of Dimension 7 (over Algebraically Closed Field and R).  PhD thesis, 1998.
  * [14] A. Gray.  Vector cross products on manifolds.  Trans. Amer. Math. Soc., 141:465–504, 1969.
  * [15] N. Hitchin.  Stable forms and special metrics.  In Global differential geometry: the mathematical legacy of Alfred Gray (Bilbao, 2000), volume 288 of Contemp. Math., pages 70–89. Amer. Math. Soc., Providence, RI, 2001.
  * [16] S. Ivanov, M. Parton, and P. Piccinni.  Locally conformal parallel $\mathrm{G}_{2}$ and ${\rm Spin}(7)$ manifolds.  Math. Res. Lett., 13(2-3):167–177, 2006.
  * [17] D. D. Joyce.  Compact manifolds with special holonomy.  Oxford Mathematical Monographs. Oxford University Press, Oxford, 2000\.
  * [18] S. Karigiannis.  Introduction to $\mathrm{G}_{2}$ geometry.  In Lectures and surveys on $\mathrm{G}_{2}$-manifolds and related topics, volume 84 of Fields Inst. Commun., pages 3–50. Springer, New York, [2020] ©2020.
  * [19] J. Lauret.  Homogeneous nilmanifolds of dimensions $3$ and $4$.  Geom. Dedicata, 68(2):145–155, 1997.
  * [20] J. Milnor.  Curvatures of left invariant metrics on Lie groups.  Advances in Math., 21(3):293–329, 1976.
  * [21] K. Nomizu.  On the cohomology of compact homogeneous spaces of nilpotent Lie groups.  Ann. of Math. (2), 59:531–538, 1954.
  * [22] F. Podestà and A. Raffero.  On the automorphism group of a closed $\rm G_{2}$-structure.  Q. J. Math., 70(1):195–200, 2019.
  * [23] The Sage Developers.  SageMath, the Sage Mathematics Software System (Version 9.3), 2021.  https://www.sagemath.org.
  * [24] E. N. Wilson.  Isometry groups on homogeneous nilmanifolds.  Geom. Dedicata, 12(3):337–346, 1982.