Extended commutator algebra for the $q$-oscillator and a related Askey-Wilson algebra

TL;DR

This paper extends the algebraic characterization of the $q$-oscillator's operators to include the operator $e^{ heta N}$ and relates it to the Askey-Wilson algebra $AW(3)$, revealing new algebraic structures.

Contribution

It introduces an extended commutator algebra for the $q$-oscillator including the operator $e^{ heta N}$ and connects it to the Askey-Wilson algebra $AW(3)$.

Findings

Extended the Lie polynomial characterization to include $e^{ heta N}$

Connected the extended algebra to the $q$-oscillator representation of $AW(3)$

Provided a new algebraic framework for the $q$-oscillator operators

Abstract

Let $q$ be a nonzero complex number that is not a root of unity. In the $q$-oscillator with commutation relation $ a a^+-qa^+ a =1$, it is known that the smallest commutator algebra of operators containing the creation and annihilation operators $a^+$ and $ a $ is the linear span of $a^+$ and $ a $, together with all operators of the form ${a^+}^l{\left[a,a^+\right]}^k$, and ${\left[a,a^+\right]}^k a ^l$, where $l$ is a nonnegative integer and $k$ is a positive integer. That is, linear combinations of operators of the form $ a ^h$ or $(a^+)^h$ with $h\geq 2$ or $h=0$ are outside the commutator algebra generated by $ a $ and $a^+$. This is a solution to the Lie polynomial characterization problem for the associative algebra generated by $a^+$ and $ a $. In this work, we extend the Lie polynomial characterization into the associative algebra $\mathcal{P}=\mathcal{P}(q)$ generated by $ a $, $a^+$, and the operator $e^{\omega N}$ for some nonzero real parameter $\omega$, where $N$ is the number operator, and we relate this to a $q$-oscillator representation of the Askey-Wilson algebra $AW(3)$.