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)/\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).

Properties

For all higher abelian homotopy groups, one has:

\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 k\geq 2.

See also

Literature

References

  1. ^ Bär 1999, page 16