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 ${\mathfrak{o}}_{q^{1/2}}(2)\oplus {\mathfrak{o}}_{q^{1/2}}(2)$ 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 $({\mathfrak{o}}_{q^{1/2}}(4),U_q(\mathfrak{su}(1,1)))$.

arXiv comment13 pages

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Cited by in corpus (6)

• The Askey-Wilson algebra and its avatars
• Higher Rank Relations for the Askey-Wilson and $q$-Bannai-Ito Algebra
• Finite-dimensional modules of the universal Askey--Wilson algebra and DAHA of type $(C_1^{\vee },C_1)$
• Howe duality and algebras of the Askey-Wilson type: an overview
• Finite-dimensional irreducible modules of the universal Askey--Wilson algebra at roots of unity
• A Howe correspondence for the algebra of the $\mathfrak{osp}(1|2)$ Clebsch-Gordan coefficients