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The spin group in dimension $n = 7$.
(Spin(7)-subgroups in Spin(8))
There are precisely 3 conjugacy classes of Spin(7)-subgroups inside Spin(8), and the triality group $Out(Spin(8))$ acts transitively on these three classes.
(Varadarajan 01, Theorem 5 on p. 6)
(G₂ is intersection of Spin(7)-subgroups of Spin(8))
The intersection inside Spin(8) of any two Spin(7)-subgroups from distinct conjugacy classes of subgroups (according to Prop. ) is the exceptional Lie group G₂, hence we have pullback squares of the form
(Varadarajan 01, Theorem 5 on p. 13)
We have the following commuting diagram of subgroup inclusions, where each square exhibits a pullback/fiber product, hence an intersection of subgroups:
Here in the bottom row we have the Lie groups
Spin(5)$\hookrightarrow$ Spin(6) $\hookrightarrow$ Spin(7) $\hookrightarrow$ Spin(8)
with their canonical subgroup-inclusions, while in the top row we have
SU(2)$\hookrightarrow$ SU(3) $\hookrightarrow$ G₂ $\hookrightarrow$ Spin(7)
and the right vertical inclusion $\iota'$ is the one of the two non-standard inclusions, according to Prop. .
The square on the right is that from Prop. .
The square in the middle is Varadarajan 01, Lemma 9 on p. 10.
The statement also follows with Onishchik 93, Table 2, p. 144:
(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
rotation groups in low dimensions:
see also
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$\,$ |
A. L. Onishchik (ed.) Lie Groups and Lie Algebras
I. A. L. Onishchik, E. B. Vinberg, Foundations of Lie Theory,
II. V. V. Gorbatsevich, A. L. Onishchik, Lie Transformation Groups
Encyclopaedia of Mathematical Sciences, Volume 20, Springer 1993
Veeravalli Varadarajan, Spin(7)-subgroups of SO(8) and Spin(8), Expositiones Mathematicae Volume 19, Issue 2, 2001, Pages 163-177 (doi:10.1016/S0723-0869(01)80027-X, pdf)
Last revised on July 18, 2024 at 11:23:46. See the history of this page for a list of all contributions to it.