∞-Lie theory (higher geometry)
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:
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 $\mathbb{R}^7$
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.
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
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).
The inclusion $G_2 \hookrightarrow GL(7)$ of def. 1 factors through the special orthogonal group
The dimension of (the manifold underlying) $G_2$ is
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
(e.g. Baez, 4.1)
We discuss various subgroups of $G_2$.
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 $\sigma_{\vert \mathbb{H}} = id_{\mathbb{H}}$.
The elementwise stabilizer group of the quaternions is SU(2):
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 $SU(2)$.
The subgroups in def. 2 sit in a short exact sequence of the form
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).
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,
classification of special holonomy manifolds by Berger's theorem:
G-structure | special holonomy | dimension | preserved differential form | |
---|---|---|---|---|
$\mathbb{C}$ | Kähler manifold | U(k) | $2k$ | Kähler forms $\omega_2$ |
Calabi-Yau manifold | SU(k) | $2k$ | ||
$\mathbb{H}$ | quaternionic Kähler manifold | Sp(k)Sp(1) | $4k$ | $\omega_4 = \omega_1\wedge \omega_1+ \omega_2\wedge \omega_2 + \omega_3\wedge \omega_3$ |
hyper-Kähler manifold | Sp(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) manifold | Spin(7) | 8 | Cayley form |
G2 manifold | G2 | $7$ | associative 3-form |
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
Cohomological properties are discussed in
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 Yang-Mills theory with $G_2$ as gauge group is in