Contents
Context
Group Theory
group theory
Classical groups
Finite groups
Group schemes
Topological groups
Lie groups
Super-Lie groups
Higher groups
Cohomology and Extensions
Related concepts
Contents
Idea
The special orthogonal group in dimension 3.
Properties
Cohomology
This is a special case of Brown 82, theorem 1.5, which is also reviewed as Rudolph-Schmidt 17, Thm. 4.2.23 with Remark 4.2.25.
Finite subgroups
Theorem
(ADE classification of finite subgroups of SO(3) and Spin(3) SU(2))
The finite subgroups of the special orthogonal group as well as the finite subgroups of the special unitary group SU(2) are, up to conjugation, given by the following classification:
Here under the double cover projection (using the exceptional isomorphism )
all the finite subgroups of except the odd-order cyclic groups are the preimages of the corresponding finite subgroups of , in that we have pullback diagrams
exhibiting the even order cyclic groups as subgroups of Spin(2), including the the minimal case of the group of order 2
as well as
exhibiting the binary dihedral groups as sitting inside the Pin(2)-subgroup of Spin(3),
but only commuting diagrams
for the odd order cyclic subgroups.
This goes back to (Klein 1884, chapter I). Full proof for is spelled out for instance in (Rees 05, theorem 11, De Visscher 11). The proof for the case of is spelled out in (Miller-Blichfeldt-Dickson 16) reviewed in (Serrano 14, section 2). The proof of the case for given the result for is spelled out in Keenan 03, theorem 4.
Irreducible representations
has an irreducible representation as the group of rotations in , whose action preserves both the dot product and cross product. This is the “defining representation”.
But something analogous happens to be true in : There is an subgroup of the exceptional Lie group (see there) for which the irreducible representation of on remains irreducible when restricted to this subgroup. preserves the dot and cross products defined there in terms of the imaginary octonions.
This -dimensional representation may also be realized as the space of harmonic cubic homogeneous polynomials on , otherwise known as the space of -orbital wavefunctions.
References
- Jason Hanson, Rotations in three, four, and five dimensions (arXiv:1103.5263)
See also
On the integral cohomology of the classifying space:
- Edgar H. Brown, The Cohomology of and with Integer Coefficients, Proceedings of the American Mathematical Society, Vol. 85, No. 2 (Jun., 1982), pp. 283-288 (jstor:2044298)
reviewed in
- Gerd Rudolph, Matthias Schmidt, around Theorem 4.2.23 of Differential Geometry and Mathematical Physics: Part II. Fibre Bundles, Topology and Gauge Fields, Theoretical and Mathematical Physics series, Springer 2017 (doi:10.1007/978-94-024-0959-8)