A higher rank extension of the Askey-Wilson Algebra

Abstract

A novel generalization of the Askey-Wilson algebra is presented and shown to be associated with coproducts in the quantum algebra $U_q(su(1,1))$. This algebra has 15 non-commuting generators given by $Q^{(A)}$, with $A\subset \{1,2,3,4\}$ and their 5 linearly independent inversions generated by the algebra automorphism $q\to q^{-1}$, $E\leftrightarrow F.$ The set of generators can be split into operators fixed under inversion, and those with an orientation under this inversion. We then show that the generators will either commute or satisfy q-commutator relations linear in the generators, with the restriction that parity operators commute with generators of opposite parity and the q-commutation relations are between those with the same parity. Finally, we give a novel algebra expression satisfied by the generators involving only the natural generators, i.e. those arising from the coupling scheme.

References in corpus (8)

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• A higher rank Racah algebra and the ${\mathbb{Z}}_2^n$ Laplace-Dunkl operator
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• Models of Quadratic Algebras Generated by Superintegrable Systems in 2D
• Racah Polynomials and Recoupling Schemes of $\mathfrak{su}(1,1)$
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Cited by in corpus (10)

• Diagonalization of the Heun-Askey-Wilson operator, Leonard pairs and the algebraic Bethe ansatz
• FRT presentation of classical Askey-Wilson algebras
• Revisiting the Askey--Wilson algebra with the universal R-matrix of $U_q(sl(2))$
• The $q$-Higgs and Askey-Wilson algebras
• Temperley-Lieb, Brauer and Racah algebras and other centralizers of su(2)
• The $q$-Bannai-Ito algebra and multivariate $(-q)$-Racah and Bannai-Ito polynomials
• Higher rank classical analogs of the Askey-Wilson algebra from the $s{l}_N$ Onsager algebra
• The dual pair $Pin(2n)\times \mathfrak{osp}(1|2)$, the Dirac equation and the Bannai-Ito algebra
• The dual pair $(U_q(\mathfrak{su}(1,1)),{\mathfrak{o}}_{q^{1/2}}(2n))$, $q$-oscillators and Askey-Wilson algebras
• A Howe correspondence for the algebra of the $\mathfrak{osp}(1|2)$ Clebsch-Gordan coefficients