# The dual pair $\big(U_q(\mathfrak{su}(1,1)),\mathfrak{o}_{q^{1/2}}(2n)\big)$, $q$-oscillators and Askey-Wilson algebras

## TL;DR

This paper explores the duality and algebraic structures of the Askey-Wilson algebra $AW(3)$ through quantum group commutants and $q$-oscillator representations, revealing new insights into their interrelations and higher-rank generalizations.

## Contribution

It demonstrates the dual realization of $AW(3)$ via different algebraic frameworks and extends the concept to higher ranks with a new dual pair approach.

## Key findings

- $AW(3)$ can be realized as a commutant in different algebraic settings.
- The duality between two realizations of $AW(3)$ is established via Howe duality.
- Higher rank extensions $AW(n)$ are constructed using dual pairs and commutants.

## Abstract

The universal Askey-Wilson algebra $AW(3)$ can be obtained as the commutant of $U_q(\mathfrak{su}(1,1))$ in $U_q(\mathfrak{su}(1,1))^{\otimes3}$. We analyze the commutant of $\mathfrak{o}_{q^{1/2}}(2)\oplus\mathfrak{o}_{q^{1/2}}(2)\oplus\mathfrak{o}_{q^{1/2}}(2)$ in $q$-oscillator representations of $\mathfrak{o}_{q^{1/2}}(6)$ and show that it also realizes $AW(3)$. These two pictures of $AW(3)$ are shown to be dual in the sense of Howe; this is made clear by highlighting the role of the intermediate Casimir elements of each members of the dual pair $\big(U_q(\mathfrak{su}(1,1)),\mathfrak{o}_{q^{1/2}}(6)\big)$. We also generalize these results. A higher rank extension of the Askey-Wilson algebra denoted $AW(n)$ can be defined as the commutant of $U_q(\mathfrak{su}(1,1))$ in $U_q(\mathfrak{su}(1,1))^{\otimes n}$ and a dual description of $AW(n)$ as the commutant of $\mathfrak{o}_{q^{1/2}}(2)^{\oplus n}$ in $q$-oscillator representations of $\mathfrak{o}_{q^{1/2}}(2n)$ is offered by calling upon the dual pair $\big(U_q(\mathfrak{su}(1,1)),\mathfrak{o}_{q^{1/2}}(2n)\big)$.