Spherical Harmonics Ylm(θ,ϕ): Positive and Negative Integer Representations of su(1,1) for l-m and l+m - H. Fakhri (Advances in High Energy Physics)
The azimuthal and magnetic quantum numbers of spherical harmonics Ylm(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 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 Ylm(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.