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rotation groups in low dimensions:
see also
is the spin group in dimension 4, the double cover of SO(4).
Let be the real vector space underlying the quaternions. Notice that Spin(3) is the group of unit quaternions under quaternion multiplication
This induces a group homomorphism
The group homomorphism (1) is a double cover and hence exhibits an isomorphism between Spin(4) and the direct product group of Spin(3) with itself:
Since the action of Spin(3) on the imaginary quaternions is the conjugation action by unit quaternions, it follows in particular, that the canonical inclusion of Spin(3) into Spin(4) is given by the diagonal morphsm with respect to the identification (2):
(e.g. Berger 87, Thm. 8.9.8, Garrett 13, §2.3)
In summary:
There is a commuting diagram of Lie groups of the form
where
in the top right we have Spin(4),
in the bottom left we have Sp(1).Sp(1)
in the bottom right we have SO(4)
the horizontal morphism assigns the conjugation action of unit quaternions, as indicated,
the right vertical morphism is the defining double cover,
the left vertical morphism is the defining quotient group-projection.
(exceptional isomorphism via Dynkin diagrams)
Under the classification of simple Lie groups via Dynkin diagrams, and via the further exceptional isomorphism Spin(3) SU(2), the exceptional isomorphism (2) corresponds to the coincidence of the D3 with the A3 diagrams, both with their central node removed:
(integral cohomology of classifying space/universal characteristic classes)
The integral cohomology ring of the classifying space of Spin(3) is freely generated from th of the first Pontryagin class:
Moreover, the integral cohomology ring of the classifying space of Spin(4) is freely generated from the first fractional Pontryagin class and the combination , where is the Euler class:
Finally, under the exceptional isomorphism (1) these classes are related by
Therefore, under the canonical diagonal inclusion (3) we have
(e.g. Čadek-Vanžura 98, Lemma 2.1)
linebreak
rotation groups in low dimensions:
see also
Marcel Berger, Section 8.9 of: Geometry I, Springer 1987 (doi:10.1007/978-3-540-93815-6)
Martin Čadek, Jiří Vanžura, On 4-fields and 4-distributions in 8-dimensional vector bundles over 8-complexes, Colloquium Mathematicum 1998, 76 (2), pp 213-228 (web)
Paul Garrett, Sporadic isogenies to orthogonal groups (July 2013) [pdf, pdf ]
Last revised on October 17, 2023 at 12:38:36. See the history of this page for a list of all contributions to it.