Spherical Harmonics Y l m ( θ , ϕ )‎ : Positive and Negative Integer Representations of su ( 1,1 )‎ for l - m and l + m

الملخص EN

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 s u ( 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 s u ( 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 s u ( 2 ) compact Lie algebra via the Y l m ( θ , ϕ ) ’s for a given l , we can also represent s u ( 1,1 ) noncompact Lie algebra by spherical harmonics for given values of l - m and l + m .