Spinh Group

In spin geometry, a spinʰ group (or quaternionic spin 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 $\mathbb$. An important application of spinʰ groups is for spinʰ structures.

Definition

The spin group $\operatorname(n)$ is a double cover of the special orthogonal group $\operatorname(n)$, hence $\mathbb_2$ acts on it with $\operatorname(n)/\Z_2\cong\operatorname(n)$. Furthermore, $\mathbb_2$ also acts on the first symplectic group $\operatorname(1)$ through the antipodal identification $y\sim -y$. The ''spinʰ group'' is then:Bär 1999, page 16

: $\operatorname^\mathrm(n) :=\left( \operatorname(n)\times\operatorname(1) \right)/\mathbb_2$

mit $(x,y)\sim(-x,-y)$. It is also denoted $\operatorname^\mathbb(n)$. Using the exceptional isomorphism $\operatorname(3) \cong\operatorname(1)$, one also has $\operatorname^\mathrm(n) =\operatorname^3(n)$ with:

: $\operatorname^k(n) :=\left( \operatorname(n)\times\operatorname(k) \right)/\mathbb_2.$

Low-dimensional examples

* $\operatorname^\mathrm(1) \cong\operatorname(1) \cong\operatorname(2)$, induced by the isomorphism $\operatorname(1) \cong\operatorname(1) \cong\mathbb_2$
* $\operatorname^\mathrm(2) \cong\operatorname(2)$, induced by the exceptional isomorphism $\operatorname(2) \cong\operatorname(1) \cong\operatorname(2)$- Since furthermore $\operatorname(3) \cong\operatorname(1) \cong\operatorname(2)$, one also has $\operatorname^\mathrm(2) \cong\operatorname^\mathrm(3)$.

Properties

For all higher abelian homotopy groups, one has:

: $\pi_k\operatorname^\mathrm(n) \cong\pi_k\operatorname(n)\times\pi_k\operatorname(1) \cong\pi_k\operatorname(n)\times\pi_k(S^3)$

for $k\geq 2$.

See also

* Spinᶜ group

Literature

* {{cite journal , author= Christian Bär , date=1999 , title=Elliptic symbols , url=https://www.researchgate.net/publication/280877898 , language=en , volume=201 , issue=1 , periodical=Mathematische Nachrichten

References

Lie groups
Differential geometry