Contents

group theory

# Contents

## Idea

The Lie group $G_2$ is one (or rather: three) of the exceptional Lie groups. One way to characterize it is as the automorphism group of the octonions as a normed algebra:

$G_2 = Aut(\mathbb{O}) \,.$

Another way to characterize it is as the stabilizer subgroup inside the general linear group $GL(7)$ of the canonical differential 3-form $\langle ,(-)\times (-) \rangle$ on the Cartesian space $\mathbb{R}^7$

$G_2 \simeq Stab_{GL(7)}(\langle -, -\times -\rangle) \,.$

As such, the group $G_2$ is a higher analog of the symplectic group (which is the group that preserves a canonical 2-form on any $\mathbb{R}^{2n}$), obtained by passing from symplectic geometry to 2-plectic geometry.

## Definition

###### Definition

On the Cartesian space $\mathbb{R}^7$ consider the associative 3-form, the constant differential 3-form $\omega \in \Omega^3(\mathbb{R}^7)$ given on tangent vectors $u,v,w \in \mathbb{R}^7$ by

$\omega(u,v,w) \coloneqq \langle u , v \times w\rangle \,,$

where

• $\langle -,-\rangle$ is the canonical bilinear form

• $(-)\times(-)$ is the cross product of vectors.

Then the group $G_2 \hookrightarrow GL(7)$ is the subgroup of the general linear group acting on $\mathbb{R}^7$ which preserves the canonical orientation and preserves this 3-form $\omega$. Equivalently, it is the subgroup preserving the orientation and the Hodge dual differential 4-form $\star \omega$.

See for instance the introduction of (Joyce).

## Properties

### Orientation

The inclusion $G_2 \hookrightarrow GL(7)$ of def. factors through the special orthogonal group

$G_2 \hookrightarrow SO(7) \hookrightarrow GL(7) \,.$

### Dimension

The dimension of (the manifold underlying) $G_2$ is

$dim(G_2) = 14 \,.$

One way to see this is via octonionic basic triples $(e_1, e_2, e_3) \in \mathbb{O}^3$ and the fact (this proposition) that these form a torsor over $G_2$, hence that the space of them has the same dimension as $G_2$:

• the space of choices for $e_1$ is the 6-sphere of imaginary unit octonions;

• given that, the space of choices for $e_2$ is a 5-sphere of imaginary unit octonions orthogonal to $e_1$;

• given that, then the space of choices for $e_3$ is the 3-sphere of imaginary unit octonions orthogonal to both $e_1$ and $e_2$.

Hence

$dim(G_2) = dim(S^6) + dim(S^5) + dim(S^3) = 14 \,.$

(e.g. Baez, 4.1)

### Cohomology

The Dwyer-Wilkerson space $G_3$ (Dwyer-Wilkerson 93) is a 2-complete H-space, in fact a finite loop space/infinity-group, such that the mod 2 cohomology ring of its classifying space/delooping is the mod 2 Dickson invariants of rank 4. As such, it is the fourth and last space in a series of infinity-groups that starts with 3 compact Lie groups:

$n=$1234
$DI(n)=$Z/2SO(3)G2G3
= Aut(C)= Aut(H)= Aut(O)

### Subgroups

We discuss various subgroups of $G_2$.

###### Definition

Write

• $G_2 = Aut(\mathbb{O})$, the automorphism group of the octonions as a normed alegbra,

• $Stab_{G_2}(\mathbb{H})$, the stabilizer subgroup of the quaternions inside the octonions, i.e. of elements $\sigma\in G_2$ such that $\sigma_{|\mathbb{H}}\colon \mathbb{H}\to \mathbb{H} \hookrightarrow\mathbb{O}$;

• $Fix_{G_2}(\mathbb{H})$ for the further subgroup of elements that fix each quaternions (the “elementwise stabilizer group”), i.e. those $\sigma$ with $\sigma_{\vert \mathbb{H}} = id_{\mathbb{H}}$.

###### Proposition

The elementwise stabilizer group of the quaternions is SU(2):

$Fix_{G_2}(\mathbb{H}) \simeq SU(2) \,.$
###### Proof

Consider octonionic basic triples $(e_1, e_2, e_3) \in \mathbb{O}^3$ and the fact (this proposition) that these form a torsor over $G_2$.

The choice of $(e_1,e_2)$ is equivalently a choice of inclusion $\mathbb{H} \hookrightarrow \mathbb{O}$. Then the remaining space of choices for $e_3$ is the 3-sphere (the space of unit imaginary octonions orthogonal to both $e_1$ and $e_2$). This carries a unit group structure, and by the torsor property this is the required subgroup of $SU(2)$.

###### Proposition

The subgroups in def. sit in a short exact sequence of the form

$\array{ 1 &=& 1 \\ \downarrow && \downarrow \\ Fix_{G_2}(\mathbb{H}) & \simeq & SU(2) \\ \downarrow && \downarrow \\ Stab_{G_2}(\mathbb{H}) & \simeq & SO(4) \\ \downarrow && \downarrow \\ Aut(\mathbb{H}) &\simeq& SO(3) \\ \downarrow && \downarrow \\ 1 &=& 1 }$

exhibiting SO(4) as a group extension of the special orthogonal group $SO(3)$ by the special unitary group $SU(2)$.

(e.g. Ferolito, section 4)

Furthermore there is a subgroup $SU(3) \hookrightarrow G_2$ whose intersection with $SO(4)$ is $U(2)$. The simple part $SU(2)$ of this intersection is a normal subgroup of $SO(4)$.

(see e.g. Miyaoka 93)

The coset space G2/SU(3) is the 6-sphere. See there for more. (from Kramer 02)

The Weyl group of $G_2$ is the dihedral group of order 12. (see e.g. Ishiguro, p. 3).

$\,$

### Supgroups

###### Proposition

(coset space of Spin(7) by G2 is 7-sphere)

Consider the canonical action of Spin(7) on the unit sphere in $\mathbb{R}^8$ (the 7-sphere),

1. This action is is transitive;

2. the stabilizer group of any point on $S^7$ is G2;

3. all G2-subgroups of Spin(7) arise this way, and are all conjugate to each other.

Hence the coset space of Spin(7) by G2 is the 7-sphere

$Spin(7)/G_2 \;\simeq\; S^7 \,.$

### Coset quotients

coset space-structures on n-spheres:

standard:
$S^{n-1} \simeq_{diff} SO(n)/SO(n-1)$this Prop.
$S^{2n-1} \simeq_{diff} SU(n)/SU(n-1)$this Prop.
$S^{4n-1} \simeq_{diff} Sp(n)/Sp(n-1)$this Prop.
exceptional:
$S^7 \simeq_{diff} Spin(7)/G_2$Spin(7)/G2 is the 7-sphere
$S^7 \simeq_{diff} Spin(6)/SU(3)$since Spin(6) $\simeq$ SU(4)
$S^7 \simeq_{diff} Spin(5)/SU(2)$since Sp(2) is Spin(5) and Sp(1) is SU(2), see Spin(5)/SU(2) is the 7-sphere
$S^6 \simeq_{diff} G_2/SU(3)$G2/SU(3) is the 6-sphere
$S^15 \simeq_{diff} Spin(9)/Spin(7)$Spin(9)/Spin(7) is the 15-sphere (from FSS 19, 3.4)

### $G$-Structure and exceptional geometry

Spin(8)-subgroups and reductions to exceptional geometry

reductionfrom spin groupto maximal subgroup
Spin(7)-structureSpin(8)Spin(7)
G2-structureSpin(7)G2
CY3-structureSpin(6)SU(3)
SU(2)-structureSpin(5)SU(2)
generalized reductionfrom Narain groupto direct product group
generalized Spin(7)-structure$Spin(8,8)$$Spin(7) \times Spin(7)$
generalized G2-structure$Spin(7,7)$$G_2 \times G_2$
generalized CY3$Spin(6,6)$$SU(3) \times SU(3)$

### Relation to higher prequantum geometry

The 3-form $\omega$ from def. we may regard as equipping $\mathbb{R}^7$ with 2-plectic structure. From this point of view $G_2$ is the linear subgroup of the 2-plectomorphism group, hence (up to the translations) the image of the Heisenberg group of $(\mathbb{R}^7, \omega)$ in the symplectomorphism group.

Or, dually, we may regard the 4-form $\star \omega$ of def. as being a 3-plectic structure and $G_2$ correspondingly as the linear part in the 3-plectomorphism group of $\mathbb{R}^7$.

• G2, F4,

E6, E7, E8, E9, E10, E11, $\cdots$

classification of special holonomy manifolds by Berger's theorem:

$\,$G-structure$\,$$\,$special holonomy$\,$$\,$dimension$\,$$\,$preserved differential form$\,$
$\,\mathbb{C}\,$$\,$Kähler manifold$\,$$\,$U(n)$\,$$\,2n\,$$\,$Kähler forms $\omega_2\,$
$\,$Calabi-Yau manifold$\,$$\,$SU(n)$\,$$\,2n\,$
$\,\mathbb{H}\,$$\,$quaternionic Kähler manifold$\,$$\,$Sp(n).Sp(1)$\,$$\,4n\,$$\,\omega_4 = \omega_1\wedge \omega_1+ \omega_2\wedge \omega_2 + \omega_3\wedge \omega_3\,$
$\,$hyper-Kähler manifold$\,$$\,$Sp(n)$\,$$\,4n\,$$\,\omega = a \omega^{(1)}_2+ b \omega^{(2)}_2 + c \omega^{(3)}_2\,$ ($a^2 + b^2 + c^2 = 1$)
$\,\mathbb{O}\,$$\,$Spin(7) manifold$\,$$\,$Spin(7)$\,$$\,$8$\,$$\,$Cayley form$\,$
$\,$G2 manifold$\,$$\,$G2$\,$$\,7\,$$\,$associative 3-form$\,$

### General

Surveys are in

• Spiro Karigiannis, What is… a $G_2$-manifold (pdf)

• Simon Salamon, A tour of exceptional geometry, (pdf)

• Wikipedia, G2 .

The definitions are reviewed for instance in

• Dominic Joyce, Compact Riemannian 7-manifolds with holonomy $G_2$, Journal of Differential Geometry vol 43, no 2 (pdf)

• Ferolito The octonions and $G_2$ (pdf)

• John Baez, section 4.1 G2, of The Octonions (arXiv:math/0105155)

• Ruben Arenas, Constructing a Matrix Representation of the Lie Group $G_2$, 2005 (pdf)

Discussion in terms of the Heisenberg group in 2-plectic geometry is in

A description of the root space decomposition of the Lie algebra $\mathfrak{g}_2$ is in

• Tathagata Basak, Root space decomposition of $\mathfrak{g}_2$ from octonions, arXiv:1708.02367

Cohomological properties are discussed in

• Younggi Choi, Homology of the gauge group of exceptional Lie group $G_2$, J. Korean Math. Soc. 45 (2008), No. 3, pp. 699–709

Discussion of subgroups includes

• Reiko Miyaoka, The linear isotropy group of $G_2/SO(4)$, the Hopf fibering and isoparametric hypersurfaces, Osaka J. Math. Volume 30, Number 2 (1993), 179-202. (Euclid)

• Kenshi Ishiguro, Classifying spaces and a subgroup of the exceptional Lie group $G_2$ pdf

• Linus Kramer, 4.27 of Homogeneous Spaces, Tits Buildings, and Isoparametric Hypersurfaces, AMS 2002

Discussion of $G_2$ as a subgroup of Spin(7):

### Applications in physics

Discussion of Yang-Mills theory with $G_2$ as gauge group is in

• Ernst-Michael Ilgenfritz, Axel Maas, Topological aspects of $G_2$ Yang-Mills theory (arXiv:1210.5963)