The Higgs and Hahn algebras from a Howe duality perspective

• Published in: Physics letters. A 2019-05, Vol.383 (14), p.1531-1535
• Main Authors: Frappat, Luc, Gaboriaud, Julien, Vinet, Luc, Vinet, Stéphane, Zhedanov, Alexei
• Format: Article
• Language: English
• Subjects: Commutant; Dimensional reduction; Hahn algebra; Higgs algebra; Howe duality; Singular oscillator
• ISSN: 0375-9601; 1873-2429
• DOI: 10.1016/j.physleta.2019.02.024

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 o(2)⊕o(2) subalgebra of o(4) in the oscillator representation of the universal algebra U(u(4)). This connection is further related to the embedding of the (discrete) Hahn algebra in U(su(1,1))⊗U(su(1,1)) in light of the dual action of the pair (o(4),su(1,1)) 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. •Hahn algebra encodes the bispectrality of the Clebsch-Gordan coefficients of su(1,1).•Higgs algebra describes the symmetries of various superintegrable models.•The isomorphic Higgs/Hahn algebras are commutants in the universal algebra of u(4).•Howe duality connects commutant picture and embedding in su(1,1)⊗su(1,1).•Higgs symmetry of 2D singular oscillator recognized from dimensional reduction.