Coordinate realizations of deformed Lie algebras with three generators

Differential realizations in coordinate space for deformed Lie algebras with three generators are obtained using bosonic creation and annihilation operators satisfying Heisenberg commutation relations. The unified treatment presented here contains as special cases all previously given coordinate realizations of $so(2,1),so(3)$ and their deformations. Applications to physical problems involving eigenvalue determination in nonrelativistic quantum mechanics are discussed.