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

Definition

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

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

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

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

Low-dimensional examples

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