The q-Higgs and Askey–Wilson algebras

Abstract

A q-analogue of the Higgs algebra, which describes the symmetry properties of the harmonic oscillator on the 2-sphere, is obtained as the commutant of the oq1/2(2)⊕oq1/2(2) subalgebra of oq1/2(4) in the q-oscillator representation of the quantized universal enveloping algebra Uq(u(4)). This q-Higgs algebra is also found as a specialization of the Askey–Wilson algebra embedded in the tensor product Uq(su(1,1))⊗Uq(su(1,1)). The connection between these two approaches is established on the basis of the Howe duality of the pair (oq1/2(4),Uq(su(1,1))).