Spinh group

In spin geometry, a spin^h group is a Lie group obtained by the spin group through twisting with the first symplectic group. H stands for the quaternions, which are denoted. An important application of spin^h groups is for spin^h structures.

Definition

The spin group is a double cover of the special orthogonal group, hence acts on it with. Furthermore, also acts on the first symplectic group through the antipodal identification. The spin^h group is then:
mit. It is also denoted. Using the exceptional isomorphism, one also has with:

Low-dimensional examples

, induced by the isomorphism
, induced by the exceptional isomorphism - Since furthermore, one also has.

Properties

For all higher abelian homotopy groups, one has:
for.

Literature

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