  The Hahn algebra encodes the bispectral properties of the eponymous
orthogonal polynomials. In the discrete case, it is isomorphic to the
polynomial algebra identified by Higgs as the symmetry algebra of the harmonic
oscillator on the $2$-sphere. These two algebras are recognized as the
commutant of a $\mathfrak{o}(2)\oplus\mathfrak{o}(2)$ subalgebra of
$\mathfrak{o}(4)$ in the oscillator representation of the universal algebra
$\mathcal{U}(\mathfrak{u}(4))$. This connection is further related to the
embedding of the (discrete) Hahn algebra in
$\mathcal{U}(\mathfrak{su}(1,1))\otimes\mathcal{U}(\mathfrak{su}(1,1))$ in
light of the dual action of the pair
$\big(\mathfrak{o}(4),\mathfrak{su}(1,1)\big)$ on the state vectors of four
harmonic oscillators. The two-dimensional singular oscillator is naturally seen
by dimensional reduction to have the Higgs algebra as its symmetry algebra.
