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arXiv · 1903.04616

The $q$-Higgs and Askey-Wilson algebras

Luc Frappat·Julien Gaboriaud·Eric Ragoucy·Luc Vinet
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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 $\mathfrak{o}_{q^{1/2}}(2) \oplus \mathfrak{o}_{q^{1/2}}(2)$ subalgebra of $\mathfrak{o}_{q^{1/2}}(4)$ in the $q$-oscillator representation of the quantized universal enveloping algebra $U_q(\mathfrak{u}(4))$. This $q$-Higgs algebra is also found as a specialization of the Askey--Wilson algebra embedded in the tensor product $U_q(\mathfrak{su}(1,1))\otimes U_q(\mathfrak{su}(1,1))$. The connection between these two approaches is established on the basis of the Howe duality of the pair $\big(\mathfrak{o}_{q^{1/2}}(4),U_q(\mathfrak{su}(1,1))\big)$.

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Luc Frappat, Julien Gaboriaud, Eric Ragoucy, Luc Vinet. 2019-03-11. The $q$-Higgs and Askey-Wilson algebras. https://doi.org/10.1016/j.nuclphysb.2019.114632

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