The Higgs and Hahn algebras from a Howe duality perspective

Abstract

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.