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. 1 factors through the special orthogonal group

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

### 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)

### 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. 2 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)

(from Kramer 02)

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

### Relation to higher prequantum geometry

The 3-form $\omega$ from def. 1 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. 1 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-structurespecial holonomydimensionpreserved differential form
$\mathbb{C}$Kähler manifoldU(k)$2k$Kähler forms $\omega_2$
Calabi-Yau manifoldSU(k)$2k$
$\mathbb{H}$quaternionic Kähler manifoldSp(k)Sp(1)$4k$$\omega_4 = \omega_1\wedge \omega_1+ \omega_2\wedge \omega_2 + \omega_3\wedge \omega_3$
hyper-Kähler manifoldSp(k)$4k$$\omega = a \omega^{(1)}_2+ b \omega^{(2)}_2 + c \omega^{(3)}_2$ ($a^2 + b^2 + c^2 = 1$)
$\mathbb{O}$Spin(7) manifoldSpin(7)8Cayley form
G2 manifoldG2$7$associative 3-form

## References

### 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

### 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)
Revised on October 30, 2017 00:04:14 by David Roberts (129.127.37.135)