The Askey-Wilson algebra AW(n) arises not only from tensor products of U_q(su(1,1)), but also as the commutant of n commuting rotations in q-oscillator representations of o_{q^{1/2}}(2n), and the two descriptions are linked by Howe duality.
The generalized Racah algebra as a commutant
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The Racah algebra $R(n)$ of rank $(n-2)$ is obtained as the commutant of the \mbox{$\mathfrak{o}(2)^{\oplus n}$} subalgebra of $\mathfrak{o}(2n)$ in oscillator representations of the universal algebra of $\mathfrak{o}(2n)$. This result is shown to be related in a Howe duality context to the definition of $R(n)$ as the algebra of Casimir operators arising in recouplings of $n$ copies of $\mathfrak{su}(1,1)$. These observations provide a natural framework to carry out the derivation by dimensional reduction of the generic superintegrable model on the $(n-1)$ sphere which is invariant under $R(n)$.
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The dual pair $\big(U_q(\mathfrak{su}(1,1)),\mathfrak{o}_{q^{1/2}}(2n)\big)$, $q$-oscillators and Askey-Wilson algebras
The Askey-Wilson algebra AW(n) arises not only from tensor products of U_q(su(1,1)), but also as the commutant of n commuting rotations in q-oscillator representations of o_{q^{1/2}}(2n), and the two descriptions are linked by Howe duality.