exceptional structures, exceptional isomorphisms
exceptional finite rotation groups:
and Kac-Moody groups:
∞-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. 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)
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:
$n=$ | 1 | 2 | 3 | 4 |
---|---|---|---|---|
$DI(n)=$ | Z/2 | SO(3) | G2 | G3 |
= Aut(C) | = Aut(H) | = Aut(O) |
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. 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)
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).
$\,$
(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),
This action is is transitive;
the stabilizer group of any point on $S^7$ is G2;
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
(e.g Varadarajan 01, Theorem 3)
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 |
see also Spin(8)-subgroups and reductions
homotopy fibers of homotopy pullbacks of classifying spaces:
(from FSS 19, 3.4)
Spin(8)-subgroups and reductions to exceptional geometry
reduction | from spin group | to maximal subgroup |
---|---|---|
Spin(7)-structure | Spin(8) | Spin(7) |
G2-structure | Spin(7) | G2 |
CY3-structure | Spin(6) | SU(3) |
SU(2)-structure | Spin(5) | SU(2) |
generalized reduction | from Narain group | to 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 | $Spin(6,6)$ | $SU(3) \times SU(3)$ |
see also: coset space structure on n-spheres
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,
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$\,$ |
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
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 $G_2$ as a subgroup of Spin(7):
Discussion of Yang-Mills theory with $G_2$ as gauge group is in
Last revised on August 25, 2019 at 10:37:26. See the history of this page for a list of all contributions to it.