A new boson realization of fusion polynomial algebras in non-Hermitian quantum mechanics: γ-deformed $su\left(2\right)$ generators, $PT$-symmetry in Fock space and Higgs algebra

A γ-deformed version of $su\left(2\right)$ algebra with non-Hermitian generators has been obtained from a bi-orthogonal system of vectors in C ². The related Jordan–Schwinger 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 which eventually leads to several multi-boson non-Hermitian Hamiltonians. Finally, the notion of global and partial $PT$-symmetry have been introduced in a typical Fock space setting. The possibility of $PT$-symmetry breaking is also discussed. The deformation parameter γ plays a crucial role in the entire formulation and non-trivially modifies the eigenfunctions under consideration. The symmetry behavior has been conceptualized in terms of a couple of toy models. (paper)