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Spin(7)

Contents

Contents

Idea

The spin group in dimension n=7n = 7.


Properties

Subgroups and Supgroups

Proposition

(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))Out(Spin(8)) acts transitively on these three classes.

(Varadarajan 01, Theorem 5 on p. 6)

Proposition

(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)

Proposition

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 ι\iota' is the one of the two non-standard inclusions, according to Prop. .

Proof

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 spaces

Proposition

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

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

  1. This action is is transitive;

  2. the stabilizer group of any point on S 7S^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 2S 7. Spin(7)/G_2 \;\simeq\; S^7 \,.

(e.g Varadarajan 01, Theorem 3)

GG-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)-structureSpin(8,8)Spin(8,8)Spin(7)×Spin(7)Spin(7) \times Spin(7)
generalized G2-structureSpin(7,7)Spin(7,7)G 2×G 2G_2 \times G_2
generalized CY3Spin(6,6)Spin(6,6)SU(3)×SU(3)SU(3) \times SU(3)

rotation groups in low dimensions:

sp. orth. groupspin grouppin group
SO(2)Spin(2)Pin(2)
SO(3)Spin(3)
SO(4)Spin(4)
SO(5)Spin(5)
Spin(6)
Spin(7)
SO(8)Spin(8)
SO(9)Spin(9)

see also

classification of special holonomy manifolds by Berger's theorem:

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

References

  • 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 March 30, 2019 at 10:04:21. See the history of this page for a list of all contributions to it.