Multiboson Expansions for the q-Oscillator and $SU(1,1)_q$

TL;DR

This paper develops a systematic expansion method to characterize hermitian representations and boson realizations of the $q$-oscillator and $su(1,1)_q$, clarifying the role of quadratic realizations in these algebraic structures.

Contribution

It introduces a unified expansion technique to classify hermitian representations and boson realizations of the $q$-oscillator and $su(1,1)_q$, highlighting the importance of quadratic realizations.

Findings

All hermitian representations obtained via expansions.

Systematic characterization of $k$-order boson realizations.

Analysis of the special role of quadratic realizations.

Abstract

All the hermitian representations of the ``symmetric" $q$-oscillator are obtained by means of expansions. The same technique is applied to characterize in a systematic way the $k$-order boson realizations of the $q$-oscillator and $su(1,1)_q$. The special role played by the quadratic realizations of $su(1,1)_q$ in terms of boson and $q$-boson operators is analysed and clarified.