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Among all special orthogonal groups , the case of is special, since in the ADE classification of simple Lie groups it corresponds to D4, which makes its representation theory enjoy triality.
(Spin(7)-subgroups in Spin(8))
There are precisely 3 conjugacy classes of Spin(7)-subgroups inside Spin(8), and the triality group acts transitively on these three classes.
(Varadarajan 01, Theorem 5 on p. 6, see also Kollross 02, Prop. 3.3 (1))
(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) Spin(6) Spin(7) Spin(8)
with their canonical subgroup-inclusions, while in the top row we have
and the right vertical inclusion is 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:
(Spin(5).Spin(3)-subgroups in SO(8))
The direct product group SO(3) SO(5) together with the groups Sp(2).Sp(1) and , with their canonical inclusions into SO(8), form 3 conjugacy classes of subgroups inside SO(8), and the triality group acts transitively on these three classes.
Similarly:
(Spin(5).Spin(3)-subgroups in Spin(8))
The groups Spin(5).Spin(3), Sp(2).Sp(1) and , with their canonical inclusions into Spin(8), form 3 conjugacy classes of subgroups inside Spin(8), and the triality group acts transitively on these three classes.
In summary we have these subgroup inclusions
permuted by triality:
graphics grabbed from FSS 19, Sec. 3.3
The homotopy groups of in low degrees:
0 |
The ordinary cohomology ring of the classifying space is:
1) with coefficients in the cyclic group of order 2:
where are the universal Stiefel-Whitney classes,
and where
2) with coefficients in the integers:
where is the first fractional Pontryagin class, is the second Pontryagin class, is the Euler class, and
is the Bockstein homomorphism.
Moreover, we have the following relations:
This is due to Quillen 71, Čadek-Vanžura 95, see Čadek-Vanžura 97, Lemma 4.1.
Consider the delooping of the triality automorphism relating Sp(2).Sp(1) with Spin(5).Spin(3) (Prop. ) on classifying spaces
Then the pullback of the universal characteristic classes of (from Prop. ) along is as follows:
In fact .
Hence, in rational cohomology:
Spin(8)-subgroups and reductions to exceptional geometry
see also: coset space structure on n-spheres
rotation groups in low dimensions:
see also
See also
On the subgroup lattice of Spin(8)
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)
Martin Čadek, Jiří Vanžura, On and -structures in 8-dimensional vector bundles, Publicacions Matemàtiques Vol. 41, No. 2 (1997), pp. 383-401 (jstor:43737249)
Megan M. Kerr, New examples of homogeneous Einstein metrics, Michigan Math. J. Volume 45, Issue 1 (1998), 115-134 (euclid:1030132086)
Andreas Kollross, Prop. 3.3 of A Classification of Hyperpolar and Cohomogeneity One Actions, Transactions of the American Mathematical Society Vol. 354, No. 2 (Feb., 2002), pp. 571-612 (jstor:2693761)
Discussion with an eye towards foundations of M-theory:
The integral cohomology of the classifying spaces and and the action of triality on these is discussed in
Alfred Gray, Paul S. Green, Sphere transitive structures and the triality automorphism, Pacific J. Math. Volume 34, Number 1 (1970), 83-96 (euclid:1102976640)
Daniel Quillen, The mod 2 cohomology rings of extra-special 2-groups and the spinor groups, Math. Ann . 194 (1971), 19
Martin Čadek, Jiří Vanžura, On the existence of 2-fields in 8-dimensional vector bundles over 8-complexes, Commentationes Mathematicae Universitatis Carolinae, vol. 36 (1995), issue 2, pp. 377-394 (dml-cz:118764)
Martin Čadek, Jiří Vanžura, Section 2 of On and -structures in 8-dimensional vector bundles, Publicacions Matemàtiques Vol. 41, No. 2 (1997), pp. 383-401 (jstor:43737249)
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