The $q$-Higgs and Askey-Wilson 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 $({\mathfrak{o}}_{q^{1/2}}(4),U_q(\mathfrak{su}(1,1)))$.

13 pages