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 advanced_tools:group_theory:u1 [2018/05/05 12:37]jakobadmin ↷ Links adapted because of a move operation advanced_tools:group_theory:u1 [2020/12/12 23:06] (current)edi [Concrete] Both sides previous revision Previous revision 2020/12/12 23:06 edi [Concrete] 2018/05/05 12:37 jakobadmin ↷ Links adapted because of a move operation2018/05/03 13:04 jakobadmin ↷ Links adapted because of a move operation2018/05/03 12:54 ↷ Links adapted because of a move operation2018/04/14 13:08 theodorekorovin 2018/04/14 13:08 theodorekorovin [Intuitive] 2018/04/14 13:07 theodorekorovin created 2020/12/12 23:06 edi [Concrete] 2018/05/05 12:37 jakobadmin ↷ Links adapted because of a move operation2018/05/03 13:04 jakobadmin ↷ Links adapted because of a move operation2018/05/03 12:54 ↷ Links adapted because of a move operation2018/04/14 13:08 theodorekorovin 2018/04/14 13:08 theodorekorovin [Intuitive] 2018/04/14 13:07 theodorekorovin created Line 15: Line 15: Take note that the tangent space of a circle is, of course, just a copy of $\mathbb{R}$ but the isomorphic space $Im \mathbb{C}$ is more convenient because its elements can be "​exponentiated"​ to give the elements $e^{i \theta}$ of $U(1)$. ​ Take note that the tangent space of a circle is, of course, just a copy of $\mathbb{R}$ but the isomorphic space $Im \mathbb{C}$ is more convenient because its elements can be "​exponentiated"​ to give the elements $e^{i \theta}$ of $U(1)$. ​ - + + ---- + + **Representations** + + The diagram below shows the defining representation of $U(1)$ in its upper branch and the conjugate representations of the same group in its lower branch. For a more detailed explanation of this diagram and more representations of $U(1)$ see [[https://​esackinger.wordpress.com/​|Fun with Symmetry]]. + + [{{ :​advanced_tools:​group_theory:​representation_theory:​u1_conj_rep.jpg?​nolink }}]  ​  ​