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We review some aspects of the Racah algebra $R(n)$, including the closurerelations, pointing out their role in superintegrability, as well as in thedescription of the symmetry algebra for several models with coalgebra symmetry.The connection includes the generic model on the $(n-1)$ sphere. We discuss analgebraic scheme of constructing Hamiltonians, integrals of the motion andsymmetry algebras. This scheme reduces to the Racah algebra $R(n)$ and themodel on the $(n-1)$ sphere only for the case of specific differential operatorrealizations. We review the method, which allows us to obtain the commutantdefined in the enveloping algebra of $\mathfrak{sl}(n)$ in the classicalsetting. The related $\mathcal{A}_3$ polynomial algebra is presented for thecase $\mathfrak{sl}(3)$. An explicit construction of the quantization of thescheme for $\mathcal{A}_3$ by symmetrization of the polynomial and thereplacement of the Berezin bracket by commutator and symmetrization of thepolynomial relations is presented. We obtain the additional quantum terms.These explicit relations are of interest not only for superintegrability, butalso for other applications in mathematical physics.

Polynomial algebras from commutants: Classical and Quantum aspects of $\mathcal{A}_3$