su(1,1)≃so(2,1) Lie algebraic extensions of the Mie-type interactions with positive constant curvature

► We extend the Mie potential to the curved spherical spaces. ► We have shown that the Mie potential in three-dimensional spherical space with constant positive curvature turns into a Rosen-Morse I potential. ► Schrödinger equation is solved for the Mie potential in three-dimensional spherical space with constant positive curvature. ► Infeld factorization is used to obtain ladder operators. ► Ladder operators for the system lead to construct the algebra so(2,1).