A New Boson realization of Fusion Polynomial Algebras in Non-Hermitian Quantum Mechanics : $\gamma$-deformed $su(2)$ generators, Partial $\mathcal{P}\mathcal{T}$-symmetry and Higgs algebra

A $\gamma$-deformed version of $su(2)$ algebra with non-hermitian generators has been obtained from a bi-orthogonal system of vectors in ${\mathbf{C}}^{\mathbf{2}}$. The related Jordan-Schwinger(J-S) map is combined with boson algebras to obtain a hierarchy of fusion polynomial algebras. This makes possible the construction of Higgs algebra of cubic polynomial type. Finally the notion of partial $\mathcal{P}\mathcal{T}$ symmetry has been introduced as characteristic feature of some operators as well as their eigenfunctions. The possibility of partial $\mathcal{P}\mathcal{T}$-symmetry breaking is also discussed. The deformation parameter $\gamma$ plays a crucial role in the entire formulation and non-trivially modifies the eigenfunctions under consideration.