Algebraic representations of polynomial angular momentum algebra and Its realizations

At first the representations of polynomial angular momentum algebra and its unitary ones are obtained appling the way to get the three-parameter 1 e algebra representations. Then find the basis which can be acted on by both Lie algebra-and the polynomial angular momentum algebra simultaneity, so under such basis the representations of deformed algebras can be shown by introducing the relation of two algebra. At last the deformed algebra's single boson operator realization and one differential realization under finite-dimensional spaces are deduced.