Retrieving "Complex Structure" from the archives

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1. U(1) Symmetry Group

   Linked via "complex structure"

   It is also closely related to the special orthogonal group $\mathrm{SO}(2)$/). Since every element of $\mathrm{SO}(2)$ can be represented by the matrix:
   $$R(\theta) = \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}$$
   The map $\theta \mapsto R(\theta)$ establishes the isomorphism $\mathrm{U}(1) \cong \mathrm{SO}(2)$. However, $\mathrm{SO}(2)$ lacks the specific complex structure necessary for certain applications in gauge theory, leading physicists to prefer the …