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 ${\displaystyle \mathbb {H} }$. An important application of spinʰ groups is for spinʰ structures.

Definition

The spin group ${\displaystyle \operatorname {Spin} (n)}$ is a double cover of the special orthogonal group ${\displaystyle \operatorname {SO} (n)}$, hence ${\displaystyle \mathbb {Z} _{2}}$ acts on it with ${\displaystyle \operatorname {Spin} (n)/\mathbb {Z} _{2}\cong \operatorname {SO} (n)}$. Furthermore, ${\displaystyle \mathbb {Z} _{2}}$ also acts on the first symplectic group ${\displaystyle \operatorname {Sp} (1)}$ through the antipodal identification ${\displaystyle y\sim -y}$. The spinʰ group is then:^[1]

${\displaystyle \operatorname {Spin} ^{\mathrm {h} }(n):=\left(\operatorname {Spin} (n)\times \operatorname {Sp} (1)\right)/\mathbb {Z} _{2}}$

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

${\displaystyle \operatorname {Spin} ^{k}(n):=\left(\operatorname {Spin} (n)\times \operatorname {Spin} (k)\right)/\mathbb {Z} _{2}.}$

Low-dimensional examples

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

Properties

For all higher abelian homotopy groups, one has:

${\displaystyle \pi _{k}\operatorname {Spin} ^{\mathrm {h} }(n)\cong \pi _{k}\operatorname {Spin} (n)\times \pi _{k}\operatorname {Sp} (1)\cong \pi _{k}\operatorname {SO} (n)\times \pi _{k}(S^{3})}$

for ${\displaystyle k\geq 2}$.

See also

• Spinᶜ group

Literature

• Christian Bär (1999). "Elliptic symbols". Mathematische Nachrichten. 201 (1).