Spinc group
In spin geometry, a spinc group is a Lie group obtained by the spin group through twisting with the first unitary group. C stands for the complex numbers, which are denoted. An important application of spinc groups is for spinc structures, which are central for Seiberg–Witten theory.
Definition
The spin group is a double cover of the special orthogonal group, hence acts on it with. Furthermore, also acts on the first unitary group through the antipodal identification. The spinc group is then:with. It is also denoted. Using the exceptional isomorphism, one also has with:
Low-dimensional examples
- , induced by the isomorphism
- , induced by the exceptional isomorphism. Since furthermore, one also has.
- , induced by the exceptional isomorphism
- is a double cover, induced by the exceptional isomorphism
Properties
for.