# A Minimal left ideal description of Geometric structures in dimensions $6$,
$7$, and $8$

Ricardo Suárez  Departament of Mathematics, UNITO, Torino, Italy
<EMAIL_ADDRESS>
###### Abstract.

In this paper we relate minimal left ideals on Clifford algebras with special
geometric structures in dimensions $6,7,$ and $8$.

##  1\. Introduction

Spinorial descriptions of special geometric structures in dimensions $6,7,$
and $8$ have been recently explored in Riemannian geometry. The spaces of
spinors that are often used are $\Delta={\mathbb{R}}^{m}$, where $m=8,8,16$,
for $SU(3)$, $G_{2}$, and $Spin(7)$ structures respectively. The goal of this
paper is to canonically identify special geometric structures in dimensions
$6,7,$ and $8$ with the left Clifford modules generated by a primitive
idempotents, i.e. a minimal left ideals for the appropriate dimensions. For
instance, in dimension $6$, the $SU(3)$ structure
$\psi_{+},\psi_{-},\omega_{0}$ defines the minimal left ideal
${\mathbb{R}}_{0,6}\cdot\frac{1}{32}(\star
q^{*}(\psi_{+}\wedge\psi_{-})+4q^{*}(\psi_{+})+4\star q^{*}(\omega_{0}))$,
where $\frac{1}{32}(\star
q^{*}(\psi_{+}\wedge\psi_{-})+4q^{*}(\psi_{+})+4\star q^{*}(\omega_{0}))$ is
the unit element in the induced minimal left ideal. In dimension $7$, the
fundamental $3$-form $\phi$ that defines the $G_{2}$ structure induces the
minimal left ideal ${\mathbb{R}}_{0,7}\cdot\frac{1}{112}(\star
q^{*}(\phi\wedge\star\phi)+7q^{*}(\phi)-7q^{*}(\star\phi)-q^{*}(\phi\wedge\star\phi))$,
where $\frac{1}{112}(\star
q^{*}(\phi\wedge\star\phi)+7q^{*}(\phi)-7q^{*}(\star\phi)-q^{*}(\phi\wedge\star\phi))$
defines the unit element in this minimal left ideal. Lastly, in dimension $8$,
the $Spin(7)$ structure induced by the $4$-form $\Omega_{0}$ defines the
minimal left ideal ${\mathbb{R}}_{0,8}\cdot\frac{1}{128}(\star
q(\Omega_{0}\wedge\Omega_{0})-8q(\Omega_{0})+q(\Omega_{0}\wedge\Omega_{0}))$.

##  2\.  Geometric structures in dimension $6,7$, and $8$

We begin with the definition of a $G$-structure on a differential manifold
$M$.

######  Definition 2.1.

Let $G$ be a closed Lie subgroup of $GL(n,{\mathbb{R}})$. A $G$-structure on
$M$ is a reduction of the structure group of the frame bundle
$\mathcal{F}(M)$, $GL(n,{\mathbb{R}})$, to $G$. That is, a $G$-structure is a
principal sub-bundle $Q$ of $\mathcal{F}(M)$ with fibre $G$. If $G$ is one of
the groups that appear in the Berger classification theorem, then $G$ is
called a special geometric structure.

Let $(M,g)$ be an oriented Riemannian spin manifold; that is, $(M,g)$ is a
Riemannian manifold with vanishing first and second Stiefel Whitney classes.
These manifolds admit at least two tensor fields, the Riemannian metric $g$
and the volume form $dV_{g}$, whose stabilizer group at any given point is the
special orthogonal group $SO(n)$. As is well known for Riemannian manifolds
the holonomy group $Hol(g)$ is strictly smaller than $SO(n)$ if and only if
there exist additional nontrivial parallel tensor fields on $M$ with respect
to the Levi-Civita connection $\nabla^{g}$. Thus $G$-structures in general can
be viewed in terms of the additional tensor fields assigned to our oriented
Riemannian manifolds. In 1955 Marcel Berger classified the possible Lie
subgroups of $O(n)$ for $n$-dimensional, simply connected manifolds with an
irreducible, non-symmetric Riemannian metric (see [14]).

######  Theorem 2.2.

Suppose that $M$ is a simply connected manifold of dimension $n$ and $g$ is an
irreducible, non-symmetric Riemannian metric on $M$. Then exactly one of the
following seven cases holds.

  1. (1)

$Hol(g)=SO(n)$.

  2. (2)

$n=2m$ with $m\geq 2$, and $Hol(g)=U(m)$ in $SO(2m)$.

  3. (3)

$n=2m$ with $m\geq 2$, and $Hol(g)=SU(m)$ in $SO(2m)$.

  4. (4)

$n=4m$ with $m\geq 2$, and $Hol(g)=Sp(m)$ in $SO(4m)$.

  5. (5)

$n=4m$ with $m\geq 2$, and $Hol(g)=Sp(m)Sp(1)$ in $SO(4m)$.

  6. (6)

$n=7$ and $Hol(g)=G_{2}$ in $SO(7)$.

  7. (7)

$n=8$ and $Hol(g)=Spin(7)$ in $SO(8)$.

In dimension $6$, an $SU(3)$ structure can be defined by a $2$-form and a
stable $3$-form which can locally be expressed as
$\omega=e^{12}+e^{34}+e^{56}$ and $\psi_{+}=e^{135}-e^{146}-e^{236}-e^{245}$.

Moreover, on a $6$-dimensional oriented Riemannian manifold, we have two
additional tensors, the almost complex structure $J$ and a complex volume form
$\Psi$ which can be expressed entirely in terms of $J$ and $\psi_{+}$:
$\Psi=\psi_{+}+i\psi_{-}$, where $\psi_{-}=J\psi_{+}$. Moreover, the
Riemannian metric can be expressed in terms of $\omega$ and $J$, given by the
tensor equation $g=\omega(\cdot,J\cdot)$. In dimension $7$, a $G_{2}$
structure on a $7$-dimensional Riemannain manifold is defined by a stable
$3$-form $\phi$ which can be locally (point-wise) defined as
$\phi=e^{123}+e^{145}-e^{257}-e^{347}+e^{167}-e^{356}+e^{246}$ with respect to
the local frame $e^{1},\ldots,e^{7}$ of $T_{p}^{*}M$. For $G_{2}$ manifolds,
the Riemannian metric and volume form are expressed in terms of the
fundamental $3$-form via the equation
$g(X,Y)dV=\frac{1}{6}(\iota_{X}\phi)\wedge(\iota_{Y}\phi)\wedge\phi$ for any
pair of vector fields $X,Y$. On dimension $8$ Riemannian manifolds, a
$Spin(7)$ structure is defined as an admissible $4$-form that is locally
expressed as:

$\Omega=e^{1234}+e^{1256}+e^{1278}+e^{1357}-e^{1368}-e^{1458}$
$-e^{1467}-e^{2358}-e^{2367}-e^{2457}+e^{2468}+e^{3456}+e^{3478}+e^{5678}.$

##  3\. Real Clifford algebras and minimal left ideals generated by primitive
idempotents

Let $V$ be a finite dimensional ${\mathbb{R}}$ vector space with quadratic
form $q$, and $V^{\otimes}=\bigoplus_{k}V^{\otimes k}$ its tensor algebra. We
define the Clifford algebra of the pair $(V,q)$ as the quotient of the tensor
algebra with the two-sided ideal $I_{q}=\langle v\otimes
v+q(x)1_{V^{\otimes}}\rangle$; that is, $C_{q}(V)=V^{\otimes}/I_{q}$. The
Clifford algebra $C_{q}(V)$ carries a natural $\mathbb{Z}_{2}$ grading, where
$C_{q}^{0}(V)$ denotes the elements of even degree, while $C_{q}^{1}(V)$
denotes the elements of odd degree. Choosing a basis for $(V,q)$, say
$e_{1},\ldots,e_{n}$, we get the following canonical basis for the Clifford
algebra generated by the following $2^{n}$ monomials:

(1) $\\{1_{V}\\}\cup\\{e_{i_{1}}\cdots
e_{i_{k}}:i_{1}<i_{2}<\cdots<i_{k},k=1,\ldots,n\\}.$

As is well known, for any quadratic space over ${\mathbb{R}}$ we have an
isomorphism with a quadratic space of the form ${\mathbb{R}}^{p,q}$, with a
signature $(p,q)$, where $p$ is the number of positive definite generators and
$q$ the number of negative definite generators. For the quadratic spaces
${\mathbb{R}}^{p,q}$ and ${\mathbb{R}}^{n}$, we denote the associated Clifford
algebras by ${\mathbb{R}}_{p,q}$ and ${\mathbb{R}}_{0,n}$ respectively.
${\mathbb{R}}_{0,n}$ is canonically isomorphic to $\bigwedge{\mathbb{R}}^{n}$
as ${\mathbb{R}}$ vector spaces. This isomorphism is achieved via the
assignment of the quantization map
$q:\bigwedge{\mathbb{R}}^{n}\rightarrow{\mathbb{R}}_{0,n}$, where
$e_{i_{1}}\wedge\cdots\wedge e_{i_{k}}\mapsto e_{i_{1}}\cdots e_{i_{k}}$. The
natural grading of both algebras is preserved via this assignment. The inverse
of this isomorphism is what is called the symbol map, which we denote
$\sigma:{\mathbb{R}}_{0,n}\rightarrow\bigwedge{\mathbb{R}}^{n}$, given by
$\sigma(e_{i_{1}}\cdots e_{i_{k}})=e_{i_{1}}\wedge\cdots\wedge e_{i_{k}}$.
These maps have natural extensions onto the spaces of exterior forms on
${\mathbb{R}}^{n}$. We denote these extensions by
$q^{*}:\bigwedge^{*}({\mathbb{R}}^{n})^{*}\rightarrow{\mathbb{R}}_{n}$ where
$q^{*}(e^{i_{i}}\wedge\cdots\wedge e^{i_{k}})=e_{i_{1}}\cdots e_{i_{k}}$, and
$\sigma^{*}:{\mathbb{R}}_{n}\rightarrow\bigwedge^{*}({\mathbb{R}}^{n})^{*}$
where $\sigma^{*}(e_{i_{1}}\cdots e_{i_{k}})=e^{i_{1}}\wedge\cdots\wedge
e^{i_{k}}$, where $e^{1},\ldots,e^{n}$ is the dual basis to the canonical
basis $e_{1},\ldots,e_{n}$; that is, $e^{i}(e_{j})=\delta_{ij}$. It is well
known that Clifford algebras are isomorphic to matrix algebras over the
division algebras ${\mathbb{R}},{\mathbb{C}},$ and $\mathbb{H}$. This
classification is given in the following theorem (see [7]).

######  Theorem 3.1.

The Clifford algebra ${\mathbb{R}}_{p,q}$, where $p+q=n$, has the following
minimal representations over ${\mathbb{R}}$, ${\mathbb{C}}$, and $\mathbb{H}$:

  1. i)

${\mathbb{R}}_{p,q}\cong M_{2^{\frac{n}{2}}}({\mathbb{R}})$ if $q-p=0,6\mod
8$.

  2. ii)

${\mathbb{R}}_{p,q}\cong M_{2^{\frac{n-1}{2}}}({\mathbb{C}})$ if $q-p=1,5\mod
8$.

  3. iii)

${\mathbb{R}}_{p,q}\cong M_{2^{\frac{n-2}{2}}}(\mathbb{H})$ if $q-p=2,4\mod
8$.

  4. iv)

${\mathbb{R}}_{p,q}\cong M_{2^{\frac{n-3}{2}}}(\mathbb{H})\oplus
M_{2^{\frac{n-3}{2}}}(\mathbb{H})$ if $q-p=3\mod 8$.

  5. v)

${\mathbb{R}}_{p,q}\cong M_{2^{\frac{n-1}{2}}}({\mathbb{R}})\oplus
M_{2^{\frac{n-1}{2}}}({\mathbb{R}})$ if $q-p=7\mod 8$.

For any semisimple algebra $A$, a minimal left ideal is of type $A\cdot e$,
where $\cdot$ is multiplication in the algebra and $e\in A$ is a primitive
idempotent. An element $e$ is primitive if it cannot be written as a sum of
orthogonal idempotents ($e=f+g$; $f^{2}=f$, $g^{2}=g$, $f\cdot g=g\cdot f=0$
see [17]). An idempotent is called minimal if it is minimal with respect to
the partial ordering $f\leq e$, which happens if and only if $ef=f=fe$ (see
[17]). A minimal left ideal $I\subset A$ is a left ideal that does not contain
any other nonzero left ideals. It is known that if $I$ is a minimal left ideal
of our algebra $A$ then either $I^{2}=0$ or $I=A\cdot e$ for some idempotent
$e\in A$ (see [17]). For our Clifford algebras, any minimal left ideal is of
the form ${\mathbb{R}}_{p,q}\cdot f$ where $f$ is a primitive idempotent.
${\mathbb{R}}_{p,q}\cdot f$ is clearly a left ${\mathbb{R}}_{p,q}$ module,
where module multiplication is given by
${\mathbb{R}}_{p,q}\times{\mathbb{R}}_{p,q}\cdot
f\rightarrow{\mathbb{R}}_{p,q}\cdot f$, via $(\phi,\psi\cdot
f)\mapsto(\phi\cdot\psi)\cdot f$, for all $\phi\in{\mathbb{R}}_{p,q}$ and
$\psi\cdot f\in{\mathbb{R}}_{p,q}\cdot f.$

######  Definition 3.2.

The minimal left ideals of the Clifford algebra ${\mathbb{R}}_{p,q}$ are
called spinor spaces, and the elements of the minimal left ideals are called
algebraic spinors.

As is consistent with spin geometry, these spinor spaces generate our spinor
representations of the appropriate dimension. The following theorem gives us
the construction and classification of minimal left ideals in
${\mathbb{R}}_{p,q}$ (see [17]).

######  Theorem 3.3.

A minimal left ideal of ${\mathbb{R}}_{p,q}$ is of type
${\mathbb{R}}_{p,q}\cdot f$ where
$f=\dfrac{1+e_{t_{1}}}{2}\cdot\cdots\cdot\dfrac{1+e_{t_{k}}}{2}$ is a
primitive idempotent in ${\mathbb{R}}_{p,q}$ and $e_{t_{1}},\ldots,e_{t_{k}}$
is a set of commuting elements of the canonical basis such that
$e_{t_{i}}^{2}=1$ for all $i=1,\ldots,k=q-r_{q-p}$; moreover, the generators
form a multiplicative group of order $2^{q-r_{q-p}}$. The numbers $r_{i}$ are
called the Randon-Hurwitz numbers, given by the recurrence formula $r_{q-p}$
subject to the conditions: $r_{0}=0,\ r_{1}=1,\ r_{2}=2,\ r_{3}=2,\ r_{j}=3$
where $4\leq j\leq 7$, $r_{i+8}=r_{i}+4$ for $i\geq 0$, $r_{-1}=-1$, and
$r_{-i}=1-i+r_{i-2}$ for $i\geq 2$.

The $k$ commuting elements generate $2^{k}$ different idempotents which yield
the decomposition of the Clifford algebra given by:

${\mathbb{R}}_{p,q}=\bigoplus_{all\ \pm\
combinations}{\mathbb{R}}_{p,q}\cdot\prod_{\alpha}\dfrac{1\pm e_{\alpha}}{2},$

where each minimal left ideal
${\mathbb{R}}_{p,q}\cdot\prod_{\alpha}\dfrac{1\pm e_{\alpha}}{2}$ is of real
dimension $2^{p+q-k}$. The algebra of endomorphisms is isomorphic to the real
matrix algebra of dimensions matching those of the above theorem; that is,
${\mathbb{R}}_{p,q}\cong End({\mathbb{R}}_{p,q}\cdot f)\cong
M_{2^{p+q-k}}({\mathbb{R}})$. Restricting our representations to the spin sub-
groups $\rho:Spin(p,q)\rightarrow Aut({\mathbb{R}}_{p,q}\cdot f)$ gives us the
usual spinor representations. In the next three sections, we properly recover
geometric structures in dimensions $6,7,$ and $8$ in terms of these minimal
left ideals.

##  4\. Recovering $SU(3)$ structures from Algebraic spinors in dimension $6$

In this section we make the association between the local description of the
$SU(3)$ structure in dimension six and its associated minimal left ideal of
${\mathbb{R}}_{0,6}$.

###  4.1. An $SU(3)$ structure recovered from a minimal left ideal in
dimension $6$

An $SU(3)$ structure in dimension $6$ is given by tensors
$\omega_{0},\psi_{\pm},J_{0}$ such that their stabilizer in ${\mathbb{R}}^{6}$
is the group $SU(3)$. For the Clifford algebra ${\mathbb{R}}_{0,6}$ we define
the primitive idempotent
$f=\left(\dfrac{1+e_{135}}{2}\right)\left(\dfrac{1-e_{146}}{2}\right)\left(\dfrac{1-e_{236}}{2}\right)=\frac{1}{8}(1+e_{135}-e_{146}-e_{236}-e_{245}-e_{3456}-e_{1234}-e_{1256})$,
where ${\mathbb{R}}_{0,6}\cdot f$ is the associated minimal left ideal. Now we
normalize the idempotent to get unit coefficients which we denote
$W=8f=1+e_{135}-e_{146}-e_{236}-e_{245}-e_{3456}-e_{1234}-e_{1256}$ in
${\mathbb{R}}_{0,6}$. Utilizing the quantization and symbol maps,
$q^{*}:\bigwedge({\mathbb{R}}^{6})^{*}\rightarrow{\mathbb{R}}_{0,6}$ and
$\sigma^{*}:{\mathbb{R}}_{0,6}\rightarrow\bigwedge({\mathbb{R}}^{6})^{*}$, as
well as the grading projection map
$\pi_{\alpha}:{\mathbb{R}}_{0,6}\rightarrow{\mathbb{R}}_{0,6}^{\alpha}$, where
$\pi_{\alpha}(x)=\langle x\rangle_{\alpha}$ is the $\alpha$ graded component,
we have $\langle W\rangle_{0}=1,\langle
W\rangle_{3}=e_{135}-e_{146}-e_{236}-e_{245},$ and $\langle
W\rangle_{4}=e_{3456}-e_{1234}-e_{1256}$. This gives us the graded
decomposition $W=\langle W\rangle_{0}+\langle W\rangle_{3}+\langle
W\rangle_{4}$. Using the Clifford Hodge dual, we have the relationship
$\star\langle W\rangle_{3}=e_{246}-e_{235}-e_{145}-e_{136}$, and $\star\langle
W\rangle_{4}=-(e_{12}+e_{56}+e_{34})$. With these relations, we recover the
$SU(3)$ structure given by $\sigma^{*}(\langle
W\rangle_{3})=\psi_{+}\in\bigwedge^{3}({\mathbb{R}}^{6})^{*}$,
$-\sigma^{*}(\star\langle
W\rangle_{3})=\psi_{-}\in\bigwedge^{3}({\mathbb{R}}^{6})^{*}$,
$-\sigma^{*}(\star\langle
W\rangle_{4})=\omega_{0}\in\bigwedge^{2}({\mathbb{R}}^{6})^{*}$.

######  Proposition 4.1.

Fix the primitive idempotent
$f=\frac{1}{8}(1+e_{135}-e_{146}-e_{236}-e_{245}-e_{3456}-e_{1234}-e_{1256})$
in ${\mathbb{R}}_{0,6}$ that defines the minimal left ideal
${\mathbb{R}}_{0,6}\cdot f$. Normalize the left ideal so that we have unit
coefficients in the sense $W=8f$. From this we recover the tensor fields that
define an $SU(3)$ structure in dimension $6$. That is, $\sigma^{*}(\langle
W\rangle_{3})=\psi_{+},-\sigma^{*}(\star\langle W\rangle_{3})=\psi_{-}$, and
$-\sigma^{*}(\star\langle W\rangle_{4})=\omega_{0}$.

###  4.2. Minimal left ideal induced from an $SU(3)$ structure

For the converse, we fix an $SU(3)$ structure in ${\mathbb{R}}^{6}$ defined by
the local tensors $\omega_{0}=e^{12}+e^{34}+e^{56}$ and
$\psi_{+}=e^{135}-e^{146}-e^{236}-e^{245}$, and $\psi_{-}=J\cdot\psi_{+}$,
where $J$ is the complex structure in ${\mathbb{R}}^{6}$. It is easy to see
that $q^{*}(\psi_{+})=e_{135}-e_{246}-e_{236}-e_{145}$,
$q^{*}(\psi_{-})=e_{136}+e_{145}+e_{235}-e_{246}$, and
$q^{*}(\omega_{0})=e_{12}+e_{56}+e_{34}$. The volume form can be expressed in
terms of $\psi_{+}$ and $\psi_{-}$ using the formula
$\psi_{+}\wedge\psi_{-}=4e^{123456}$; hence the volume element of the Clifford
algebra ${\mathbb{R}}_{0,6}$ can be expressed by the formula
$\frac{1}{4}q^{*}(\psi_{+}\wedge\psi_{-})=e_{123456}$. Now via the Clifford
Hodge product we have $\frac{1}{4}\star q^{*}(\psi_{+}\wedge\psi_{-})=1$.
Hence we have the formula $\frac{1}{32}(\star
q^{*}(\psi_{+}\wedge\psi_{-})+4q^{*}(\psi_{+})+4\star
q^{*}(\omega_{0}))=\frac{1}{8}(1+e_{135}-e_{146}-e_{236}-e_{245}-e_{3456}-e_{1234}-e_{1256})=\left(\dfrac{1+e_{135}}{2}\right)\left(\dfrac{1-e_{146}}{2}\right)\left(\dfrac{1-e_{236}}{2}\right)$,
where $e_{135},e_{146},e_{236}$ are commuting positive definite elements in
the canonical basis. Hence we have a primitive idempotent, establishing the
following:

######  Proposition 4.2.

The $SU(3)$ structure $\omega_{0},\psi_{+},\psi_{-}$ in ${\mathbb{R}}^{6}$
determines the primitive idempotent $f=\frac{1}{32}(\star
q^{*}(\psi_{+}\wedge\psi_{-})+4q^{*}(\psi_{+})+4\star q^{*}(\omega_{0}))$ in
the Clifford algebra ${\mathbb{R}}_{0,6}$, and the resulting minimal left
ideal is ${\mathbb{R}}_{0,6}\cdot\frac{1}{32}(\star
q^{*}(\psi_{+}\wedge\psi_{-})+4q^{*}(\psi_{+})+4\star q^{*}(\omega_{0}))$.

Definitionally, elements of our minimal left ideal
$\sigma\in{\mathbb{R}}_{0,6}\cdot\frac{1}{32}(\star
q^{*}(\psi_{+}\wedge\psi_{-})+4q^{*}(\psi_{+})+4\star q^{*}(\omega_{0}))$ are
algebraic spinors generated by $SU(3)$ structures. Although it is enough to
have this canonical identification, from a well defined $SU(3)$ structure
$\psi_{+},\psi_{-},\omega_{0}$ we can always generate a primitive idempotent
and hence a minimal left ideal in this canonical manner.

Now the primitive idempotent, $f=\frac{1}{32}(\star
q^{*}(\psi_{+}\wedge\psi_{-})+4q^{*}(\psi_{+})+4\star q^{*}(\omega_{0}))$,
represents the identity class in the canonical projection
$\phi\mapsto\phi\cdot f$. Moreover, we have the ${\mathbb{R}}$ basis
$f,e_{2}f,e_{3}f,e_{5}f,e_{23}f,e_{25}f,e_{35}f,e_{235}f$ in
${\mathbb{R}}_{0,6}\cdot f$, where each basis element is a unit spinor in the
module.

##  5\. Algebraic spinors in dimension $7$

In dimension $7$, a $G_{2}$ structure is a positive $3$-form whose local
tensor in ${\mathbb{R}}^{7}$ is given by
$\phi=e^{123}+e^{145}+e^{167}+e^{246}-e^{257}-e^{347}-e^{356}\in\bigwedge^{3}({\mathbb{R}}^{7})^{*}$,
and whose fundamental $4$-form with respect to the Hodge star operator is
given by
$\star\phi_{0}=e^{4567}+e^{2367}+e^{2345}+e^{1357}-e^{1346}-e^{1256}-e^{1247}\in\bigwedge^{4}({\mathbb{R}}^{7})^{*}$.

###  5.1.  $G_{2}$ structure recovered from a minimal left ideal.

We begin by fixing the primitive idempotent
$f=\frac{1}{16}(1+e_{123})(1+e_{145})(1-e_{257})(1+e_{167})$, which generates
the minimal left ideal ${\mathbb{R}}_{0,7}\cdot f$. We normalize the
idempotent to obtain unit scalars via
$W=16f=1+e_{123}+e_{145}-e_{2345}-e_{257}-e_{1357}+e_{1247}-e_{347}+e_{167}-e_{2367}-e_{4567}-e_{1234567}+e_{1256}-e_{356}-e_{246}-e_{1346}$.
Using the symbol and projection maps, we have $\langle
W\rangle_{3}=e_{123}+e_{145}+e_{167}+e_{246}-e_{257}-e_{347}-e_{356}$, and
$\langle
W\rangle_{4}=e_{2367}-e_{4567}+e_{1346}+e_{1256}-e_{2345}-e_{1357}+e_{1247}$.
In terms of the projection decomposition, we write $W=1+\langle
W\rangle_{3}+\langle W\rangle_{4}-e_{1234567}$, where $e_{1234567}$ is the
volume element in ${\mathbb{R}}_{0,7}$. Using the Clifford Hodge dual, we have
$\star\langle W\rangle_{3}=\langle W\rangle_{4}$ and $\star e_{1234567}=1$,
and thus we have $W=\star e_{1234567}+\langle W\rangle_{3}+\star\langle
W\rangle_{3}-e_{1234567}$. Hence, using the extended symbol map, we have
$\sigma^{*}(\langle
W\rangle_{3})=\phi_{0}\in\bigwedge^{3}({\mathbb{R}}^{7})^{*}$, which is the
desired $G_{2}$ structure in ${\mathbb{R}}^{7}$. The associated $4$-form that
comes with a $G_{2}$ structure is given by $-\sigma^{*}(\langle
W\rangle_{4})=\star\phi_{0}$. From this we have the following.

######  Proposition 5.1.

Fix the primitive idempotent
$f=\frac{1}{16}(1+e_{123})(1+e_{145})(1-e_{257})(1+e_{167})$, with
${\mathbb{R}}_{0,7}\cdot f$ the associated minimal left ideal. We can then
obtain a $G_{2}$ structure in ${\mathbb{R}}^{7}$ generated from the normalized
tensor $W$ via $\sigma^{*}(\langle W\rangle_{3})=\phi_{0}$, and the associated
$4$-form is then given by $-\sigma^{*}(\langle W\rangle_{4})=\star\phi_{0}$.

###  5.2. Algebraic spinors generated by $G_{2}$ structures in dimension $7$

Fix a $G_{2}$ structure
$\phi=e^{123}+e^{145}+e^{167}+e^{246}-e^{257}-e^{347}-e^{356}\in\bigwedge^{3}({\mathbb{R}}^{7})^{*}$.
Using the quantization map, we have
$q^{*}(\phi)=e_{123}+e_{145}+e_{167}+e_{246}-e_{257}-e_{347}-e_{356}$ and
$q^{*}(\star\phi)=-e_{2367}+e_{4567}-e_{1346}-e_{1256}+e_{2345}+e_{1357}-e_{1247}$.
The volume form in ${\mathbb{R}}^{7}$ expressed in terms of the $G_{2}$
structure is given by the formula $\phi\wedge\star\phi=7e^{1234567}$, and thus
we have on ${\mathbb{R}}_{0,7}$ the equation
$q^{*}(\phi\wedge\star\phi)=7e_{1234567}$. Using the Clifford Hodge star
operator, we have $\star q^{*}(\phi\wedge\star\phi)=7$. Putting this all
together, we get the following primitive idempotent element in
${\mathbb{R}}_{0,7}$ induced by the $G_{2}$ structure:

$f_{\phi}=\frac{1}{112}(\star
q^{*}(\phi\wedge\star\phi)+7q^{*}(\phi)-7q^{*}(\star\phi)-q^{*}(\phi\wedge\star\phi))=\frac{1}{16}(1+e_{123})(1+e_{145})(1-e_{257})(1+e_{167}).$

######  Proposition 5.2.

Fix the local $G_{2}$ structure
$\phi=e^{123}+e^{145}+e^{167}+e^{246}-e^{257}-e^{347}-e^{356}$ in
${\mathbb{R}}^{7}$. The $3$-form $\phi$ then determines the primitive
idempotent $f_{\phi}=\frac{1}{112}(\star
q^{*}(\phi\wedge\star\phi)+7q^{*}(\phi)-7q^{*}(\star\phi)-q^{*}(\phi\wedge\star\phi))$
in the Clifford algebra ${\mathbb{R}}_{0,7}$, with the resulting minimal left
ideal ${\mathbb{R}}_{7}\cdot\frac{1}{112}(\star
q^{*}(\phi\wedge\star\phi)+7q^{*}(\phi)-7q^{*}(\star\phi)-q^{*}(\phi\wedge\star\phi))=:{\mathbb{R}}_{7}\cdot
f_{\phi}$.

Elements of the minimal left ideal ${\mathbb{R}}_{0,7}\cdot f_{\phi}$ are
algebraic spinors generated by the $G_{2}$ structure. Although it is enough to
have this canonical identification, from a well defined $G_{2}$ structure
$\phi$ we can always generate a primitive idempotent and hence a minimal left
ideal in this canonical manner. As vector spaces we have
${\mathbb{R}}_{0,7}\cdot f_{\phi}\cong{\mathbb{R}}^{8}$, with the basis of
equivalence classes given by
$f_{\phi},e_{1}f_{\phi},e_{2}f_{\phi},e_{3}f_{\phi},e_{4}f_{\phi},e_{5}f_{\phi},e_{6}f_{\phi},e_{7}f_{\phi}$.

##  6\. Algebraic spinors in dimension $8$

In dimension $8$, a $Spin(7)$ structure is defined by the model tensor
$\Omega_{0}=e^{1234}+e^{1256}+e^{1278}+e^{1357}-e^{1368}-e^{1458}-e^{1467}-e^{2358}-e^{2367}-e^{2457}+e^{2468}+e^{3456}+e^{3478}+e^{5678}\in\wedge^{4}({\mathbb{R}}^{8})^{*}.$
Using the quantization map, we have the following:
$q^{*}(\Omega_{0})=e_{1234}+e_{1256}+e_{1278}+e_{1357}-e_{1368}-e_{1458}-e_{1467}-e_{2358}-e_{2367}-e_{2457}+e_{2468}+e_{3456}+e_{3478}+e_{5678},$
$q^{*}(\Omega_{0}\wedge\star\Omega_{0})=8e_{12345678}$, and $\star
q^{*}(\Omega_{0}\wedge\star\Omega_{0})=8$. Now the formula

$f_{\Omega}=\frac{1}{128}(\star
q(\Omega_{0}\wedge\Omega)-8q(\Omega_{0})+q(\Omega_{0}\wedge\Omega_{0}))$

is a primitive idempotent in ${\mathbb{R}}_{0,8}$, as it factors out as
$f_{\Omega}=\frac{1}{16}(1-e_{1234})(1-e_{1256})(1-e_{1278})(1-e_{1357})$.
Thus ${\mathbb{R}}_{0,8}\cdot\frac{1}{128}(\star
q(\Omega_{0}\wedge\Omega)-8q(\Omega_{0})+q(\Omega_{0}\wedge\Omega_{0}))$ is
the minimal left ideal induced by our $Spin(7)$ structure. Moreover,
${\mathbb{R}}_{0,8}\cdot\frac{1}{128}(\star
q(\Omega_{0}\wedge\Omega)-8q(\Omega_{0})+q(\Omega_{0}\wedge\Omega_{0}))\cong\Delta_{8}={\mathbb{R}}^{16}$.
We summarize this with the following proposition.

######  Proposition 6.1.

The $Spin(7)$ structure $\Omega_{0}$ in ${\mathbb{R}}^{8}$ determines the
primitive idempotent $f_{\Omega}=\frac{1}{128}(\star
q(\Omega_{0}\wedge\Omega)-8q(\Omega_{0})+q(\Omega_{0}\wedge\Omega_{0}))$ in
the Clifford algebra ${\mathbb{R}}_{0,8}$, and the resulting minimal left
ideal is ${\mathbb{R}}_{0,8}\cdot\frac{1}{128}(\star
q(\Omega_{0}\wedge\Omega)-8q(\Omega_{0})+q(\Omega_{0}\wedge\Omega_{0}))$.

Conversely, for ${\mathbb{R}}_{0,8}$ we have four commuting generators from
our canonical basis $e_{1234},e_{1256},e_{1278},e_{1357}$, in which the
idempotent $f=\frac{1}{16}(1-e_{1234})(1-e_{1256})(1-e_{1278})(1-e_{1357})$
defines the spinor space ${\mathbb{R}}_{0,8}\cdot f$ isomorphic to
$\Delta_{8}={\mathbb{R}}^{16}$. The normalized tensor,
$W=16f=1-(e_{1234}+e_{1256}+e_{1278}+e_{1357}-e_{1368}-e_{1458}-e_{1467}-e_{2358}-e_{2367}-e_{2457}+e_{2468}+e_{3456}+e_{3478}+e_{5678})+e_{12345678}$,
decomposes as $W=\star e_{12345678}-\langle W\rangle_{4}+e_{12345678}$. Hence
we define $\sigma^{*}(\langle W\rangle_{4})=\Omega_{0}$, and
$\sigma^{*}(\star\langle W\rangle_{4})=\star\Omega_{0}=\Omega_{0}$. This
results in the following proposition:

######  Proposition 6.2.

Fix the primitive idempotent
$f=\frac{1}{16}(1-e_{1234})(1-e_{1256})(1-e_{1278})(1-e_{1357})$ such that
${\mathbb{R}}_{0,8}\cdot f$ is the minimal left ideal. We can associate a
$Spin(7)$ structure in ${\mathbb{R}}^{8}$ generated from the normalized tensor
$W$ via $\sigma^{*}(\langle W\rangle_{4})=\sigma^{*}(\star\langle
W\rangle_{4})=\Omega_{0}$.

##  7\. Primitive idempotents in dimension $6$ to $G_{2}$ structures in
dimension $7$

We conclude this paper by relating the constructions in dimension $6$ and
dimension $7$ by viewing
${\mathbb{R}}^{7}={\mathbb{R}}^{6}\oplus{\mathbb{R}}$, where the orthogonal
dimension is given by the basis vector $e_{7}$. We define a generic $SU(3)$
structure given in the ${\mathbb{R}}^{6}$ component by
$\omega_{0}=e^{12}+e^{34}+e^{56}$, $\psi_{+}=e^{135}-e^{146}-e^{236}-e^{245}$,
and $\psi_{-}=e^{136}+e^{145}+e^{235}-e^{246}$. For the associated Clifford
algebra ${\mathbb{R}}_{0,6}$ we have the associated primitive idempotent
$f=\left(\dfrac{1+e_{135}}{2}\right)\left(\dfrac{1-e_{146}}{2}\right)\left(\dfrac{1-e_{236}}{2}\right)=\frac{1}{8}(1+e_{135}-e_{146}-e_{236}-e_{245}-e_{3456}-e_{1234}-e_{1256})$,
with ${\mathbb{R}}_{0,6}\cdot f$ being the associated minimal left ideal. As
we saw above from the normalized idempotent $W$ in dimension $6$, we recover
the $SU(3)$ structure via $\sigma^{*}(\langle
W\rangle_{3})=\psi_{+},\sigma^{*}(\star\langle W\rangle_{3})=\psi_{-}$, and
$\sigma^{*}(\star\langle W\rangle_{4})=\omega_{0}$. Now from the induced
$SU(3)$ structure in dimension $6$, we can recover a $G_{2}$ structure in
dimension $7$ via $\phi_{0}=\sigma^{*}(\star\langle W\rangle_{4})\wedge
e^{7}+\sigma^{*}(\langle W\rangle_{3})$. Conversely, starting with the $SU(3)$
we have the primitive idempotent $\frac{1}{32}(\star
q^{*}(\psi_{+}\wedge\psi_{-})+4q^{*}(\psi_{+})+4\star q^{*}(\omega_{0}))$,
which gives us the minimal left ideal
${\mathbb{R}}_{0,6}\cdot\frac{1}{32}(\star
q^{*}(\psi_{+}\wedge\psi_{-})+4q^{*}(\psi_{+})+4\star q^{*}(\omega_{0}))$. Now
the normalized idempotent with unit scalars $W$ in this formulation is given
by $W=\star q^{*}(\psi_{+}\wedge\psi_{-})+4q^{*}(\psi_{+})+4\star
q^{*}(\omega_{0})$, and hence we define the primitive idempotent in dimension
$7$ as follows: $f_{\phi}=\frac{1}{112}(\star q^{*}(\sigma^{*}(\star\langle
W\rangle_{4})\wedge e^{7}+\sigma^{*}(\langle
W\rangle_{3})\wedge\star\sigma^{*}(\star\langle W\rangle_{4})\wedge
e^{7}+\sigma^{*}(\langle W\rangle_{3}))+7q^{*}(\sigma^{*}(\star\langle
W\rangle_{4})\wedge e^{7}+\sigma^{*}(\langle
W\rangle_{3}))-7q^{*}(\star\sigma^{*}(\star\langle W\rangle_{4})\wedge
e^{7}+\sigma^{*}(\langle W\rangle_{3}))-q^{*}(\sigma^{*}(\star\langle
W\rangle_{4})\wedge e^{7}+\sigma^{*}(\langle
W\rangle_{3})\wedge\star\sigma^{*}(\star\langle W\rangle_{4})\wedge
e^{7}+\sigma^{*}(\langle W\rangle_{3}))).$ Thus ${\mathbb{R}}_{0,7}\cdot
f_{\phi}$ is the spinor module induced from the $SU(3)$ structures recovered
from the primitive idempotents in dimension $6$.

##  8\. Future research

With the correspondences established above, we can construct algebraic spinor
bundles from the induced spinor spaces and provide classification equations
for $G_{2}$, $SU(3)$, and $Spin(7)$ manifolds in terms of algebraic spinors.

##  9\. Acknowledgments

I would like to thank my advisor, Dr. Ivona Grzegorczyk, and my supervisor at
UNITO, Dr. Anna Fino, as well as UNITO for financing my research and providing
me guidance in this topic.

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