Diffrence between U(1) and Spin(1)?

[Hymne]

Ah nice! Is it always the case the the elements of the Spin(2n) group are isomorphic to the elements in the U(n) group, and the diffrence between them are only seen in how they act on our vectorspace? The whole thing seems rather confusing..

 

[fzero]

Actually in this case there is an accidental isomorphism. Since $$SO(2)=U(1)$$ is simply connected, it is its own covering space, so $$\mathrm{Spin}(2)=U(1)$$ and there is no double cover. For $$\mathrm{Spin}(n>2)$$, there is a true double cover of $$SO(n>2)$$.

 

[Hymne]

Thanks, really nice!

Can you confirm that the only mathematical diffrence between these groups is actually the way that they act on our vector space?

 

[fzero]

Thanks, really nice!

Can you confirm that the only mathematical diffrence between these groups is actually the way that they act on our vector space?

No, there is no isomorphism between Spin(2n) and U(n), except for n=1. You can verify this by just counting the dimensions of these groups. For low dimensions there are other accidental isomorphisms (see http://en.wikipedia.org/wiki/Spin_group#Accidental_isomorphisms ), but in higher dimensions there are none.