Irreducibility and Compositeness in q-Deformed Harmonic Oscillator Algebras

q-Deformed harmonic oscillator algebra for real and root of unity values of the deformation parameter is discussed by using an extension of the number concept proposed by Gauss, namely the Q-numbers. A study of the reducibility of the Fock space representation which explores the properties of the Gauss polynomials is presented. When the deformation parameter is a root of unity, an interesting result comes out in the form of a reducibility scheme for the space representation which is based on the classification of the primitive or non-primitive character of the deformation parameter. An application is carried out for a q-deformed harmonic oscillator Hamiltonian, to which the reducibility scheme is explicitly applied. For finite-dimensional spaces associated to non-primitive roots of unity the compositeness of the