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

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Discussion Overview

The discussion revolves around the differences between the groups U(1) and Spin(1), as well as their higher-dimensional counterparts, U(n) and Spin(2n). Participants explore the nature of these groups, their isomorphisms, and how they act on vector spaces, with a focus on mathematical properties and relationships.

Discussion Character

  • Technical explanation
  • Debate/contested

Main Points Raised

  • Some participants propose that elements of the Spin(2n) group are isomorphic to elements in the U(n) group, with differences primarily in their action on vector spaces.
  • Others argue that, specifically for n=1, Spin(2) is isomorphic to U(1), but for n>1, there is no isomorphism between Spin(2n) and U(n), which can be verified by counting dimensions.
  • A later reply mentions that while there are accidental isomorphisms in low dimensions, there are none in higher dimensions.

Areas of Agreement / Disagreement

Participants do not reach a consensus; there are competing views regarding the isomorphisms and the nature of the differences between the groups.

Contextual Notes

Participants reference the concept of accidental isomorphisms and the implications of group dimension, indicating that the discussion may depend on specific mathematical definitions and contexts.

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

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