Spherical Harmonics $Y_{l}^{m}(θ,ϕ)$: Positive and Negative Integer Representations of su(1,1) for l-m and l+m

The azimuthal and magnetic quantum numbers of spherical harmonics $Y_{l}^{m}(θ,ϕ)$ describe quantization corresponding to the magnitude and $z$-component of angular momentum operator in the framework of realization of $su(2)$ Lie algebra symmetry. The azimuthal quantum number $l$ allocates to itself an additional ladder symmetry by the operators which are written in terms of $l$. Here, it is shown that simultaneous realization of the both symmetries inherits the positive and negative $(l-m)$- and $(l+m)$-integer discrete irreducible representations for $su(1,1)$ Lie algebra via the spherical harmonics on the sphere as a compact manifold. So, in addition to realizing the unitary irreducible representation of $su(2)$ compact Lie algebra via the $Y_{l}^{m}(θ,ϕ)$'s for a given $l$, we can also represent $su(1,1)$ noncompact Lie algebra by spherical harmonics for given values of $l-m$ and $l+m$.

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