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1. U(1) Symmetry Group

   Linked via "spin systems"

   The Lie algebra associated with $\mathrm{U}(1)$, denoted $\mathfrak{u}(1)$, is one-dimensional and is spanned by the generator $T$ corresponding to infinitesimal transformations:
   $$U(\epsilon) = e^{i\epsilon T} \approx 1 + i\epsilon T$$
   For $\mathrm{U}(1)$, the generator $T$ is simply the identity multiplied by a real scaling factor, often normalized such that $T=1/2$ in specific contexts (like spin systems), or $T=1$ when relating directly to the imaginary unit $i$ in the exponent. The…