# The $q$-Higgs and Askey-Wilson algebras

## TL;DR

This paper introduces a $q$-analogue of the Higgs algebra related to harmonic oscillators on the 2-sphere, connecting it to the Askey--Wilson algebra through Howe duality in quantum algebra representations.

## Contribution

It constructs a $q$-Higgs algebra as a commutant in quantum groups and links it to the Askey--Wilson algebra via Howe duality, revealing new algebraic structures.

## Key findings

- $q$-Higgs algebra as a commutant in $U_q(rak u(4))$
- Connection established between $q$-Higgs and Askey--Wilson algebras
- Representation theory insights into quantum symmetry algebras

## Abstract

A $q$-analogue of the Higgs algebra, which describes the symmetry properties of the harmonic oscillator on the $2$-sphere, is obtained as the commutant of the $\mathfrak{o}_{q^{1/2}}(2) \oplus \mathfrak{o}_{q^{1/2}}(2)$ subalgebra of $\mathfrak{o}_{q^{1/2}}(4)$ in the $q$-oscillator representation of the quantized universal enveloping algebra $U_q(\mathfrak{u}(4))$. This $q$-Higgs algebra is also found as a specialization of the Askey--Wilson algebra embedded in the tensor product $U_q(\mathfrak{su}(1,1))\otimes U_q(\mathfrak{su}(1,1))$. The connection between these two approaches is established on the basis of the Howe duality of the pair $\big(\mathfrak{o}_{q^{1/2}}(4),U_q(\mathfrak{su}(1,1))\big)$.