Spinh group

Short description: Twisted spin group

In spin geometry, a spin^h 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 $\mathbb{ℍ}$. An important application of spin^h groups is for spin^h structures.

Definition

The spin group $\mathrm{Spin}(n)$ is a double cover of the special orthogonal group $\mathrm{SO}(n)$, hence ${\mathbb{\mathbb{Z}}}_2$ acts on it with $\mathrm{Spin}(n)/{\mathbb{\mathbb{Z}}}_2\cong \mathrm{SO}(n)$. Furthermore, ${\mathbb{\mathbb{Z}}}_2$ also acts on the first symplectic group $\mathrm{Sp}(1)$ through the antipodal identification $y\sim -y$. The spin^h group is then:

${\mathrm{Spin}}^{\mathrm{h}}(n):=(\mathrm{Spin}(n)\times \mathrm{Sp}(1))/{\mathbb{\mathbb{Z}}}_2$

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

${\mathrm{Spin}}^k(n):=(\mathrm{Spin}(n)\times \mathrm{Spin}(k))/{\mathbb{\mathbb{Z}}}_2.$

Low-dimensional examples

• ${\mathrm{Spin}}^{\mathrm{h}}(1)\cong \mathrm{Sp}(1)\cong \mathrm{SU}(2)$, induced by the isomorphism $\mathrm{Spin}(1)\cong \mathrm{O}(1)\cong {\mathbb{\mathbb{Z}}}_2$
• ${\mathrm{Spin}}^{\mathrm{h}}(2)\cong \mathrm{U}(2)$, induced by the exceptional isomorphism $\mathrm{Spin}(2)\cong \mathrm{U}(1)\cong \mathrm{SO}(2)$- Since furthermore $\mathrm{Spin}(3)\cong \mathrm{Sp}(1)\cong \mathrm{SU}(2)$, one also has ${\mathrm{Spin}}^{\mathrm{h}}(2)\cong {\mathrm{Spin}}^{\mathrm{c}}(3)$.

Properties

For all higher abelian homotopy groups, one has:

${\pi }_k{\mathrm{Spin}}^{\mathrm{h}}(n)\cong {\pi }_k\mathrm{Spin}(n)\times {\pi }_k\mathrm{Sp}(1)\cong {\pi }_k\mathrm{SO}(n)\times {\pi }_k(S^3)$

for $k\ge 2$.

See also

• Spin^c group

Literature

• Christian Bär (1999). "Elliptic symbols" (in en). Mathematische Nachrichten 201 (1). https://www.researchgate.net/publication/280877898.