The dual pair (Uq(su(1,1)),oq1/2(2n)), <i>q</i>-oscillators, and Askey-Wilson algebras
The dual pair (Uq(su(1,1)),oq1/2(2n)), <i>q</i>-oscillators, and Askey-Wilson algebras
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 …