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  1. Fractional Symmetry Algebra

    Linked via "imaginary unit"

    The most cited, yet least understood, relationship is the 'Hessler-Reynold' bracket:
    $$[\mathcal{R}{1/2}, \mathcal{F}{1/4}] = i \cdot \Gamma$$
    Here, $i$ is the imaginary unit, and $\Gamma$ is the 'Non-Commutative Torsion Constant,' which is empirically found to be proportional to the environmental permittivity factor ($\epsilon_r$) of the vacuum in which the symmetry is observed [4].
    Classification of FSA Groups

  2. U(1) Symmetry Group

    Linked via "imaginary unit"

    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…