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The Higher Rank q-Deformed Bannai-Ito and Askey-Wilson Algebra

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The q-deformed Bannai-Ito algebra was recently constructed in the threefold tensor product of the quantum superalgebra \(\mathfrak {osp}_q(1\vert 2)\). It turned out to be isomorphic to the Askey-Wilson algebra. In the present paper these results will be extended to higher rank. The rank \(n-2\)q-Bannai-Ito algebra \(\mathcal {A}_n^q\), which by the established isomorphism also yields a higher rank version of the Askey-Wilson algebra, is constructed in the n-fold tensor product of \(\mathfrak {osp}_q(1\vert 2)\). An explicit realization in terms of q-shift operators and reflections is proposed, which will be called the \(\mathbb {Z}_2^n\)q-Dirac–Dunkl model. The algebra \(\mathcal {A}_n^q\) is shown to arise as the symmetry algebra of the constructed \(\mathbb {Z}_2^n\)q-Dirac–Dunkl operator and to act irreducibly on modules of its polynomial null-solutions. An explicit basis for these modules is obtained using a q-deformed \(\mathbf {CK}\)-extension and Fischer decomposition.

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Acknowledgements

This work was supported by the Research Foundation Flanders (FWO) under Grant EOS 30889451 and G.0116.13N. We wish to thank Toon Baeyens for writing Java code that was used to verify some of the formulae in the paper. We would also like to thank the referee for valuable comments and suggestions.

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  1. Department of Electronics and Information Systems, Faculty of Engineering and Architecture, Ghent University, Building S8, Krijgslaan 281, 9000, Gent, Belgium

    Hendrik De Bie, Hadewijch De Clercq & Wouter van de Vijver

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  1. Hendrik De Bie
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Appendix A: Expressions in the fourfold tensor product

Appendix A: Expressions in the fourfold tensor product

In this appendix we will give the explicit expression for the element \(\Gamma _A^q \in \mathfrak {osp}_q(1\vert 2)^{\otimes 4}\) for some of the non-trivial sets \(A\subset \{1,2,3,4\}\). These can be obtained using the definition of the coproduct \(\Delta \) and the coaction \(\tau \) as in (6), (8) and (17).

$$\begin{aligned} \Gamma _{\{1,3\}}^q =&\frac{-q^{1/2}}{q - q^{-1}} K^{2} P \otimes 1 \otimes K^{2} P \otimes 1 - (q-q^{-1}) A_- A_+ P \otimes A_+ K \otimes A_- K \otimes 1 \\&+ A_- A_+ P \otimes 1 \otimes K^{2} P \otimes 1 -q^{1/2} A_- K^{-1} P \otimes K^{-2} P \otimes A_+ K \otimes 1 \\&- q^{-1/2} A_- K^{-1} P \otimes A_+ K^{-1} P \otimes K^{-2} P \otimes 1\\&+ q^{-1/2} A_+ K^{-1} P \otimes K^{2} P \otimes A_- K \otimes 1 + q^{-1/2} A_- K^{-1} P \\&\otimes A_+ K^{-1} P \otimes K^{2} P \otimes 1 \\&- (q-q^{-1}) A_- K^{-1} P \otimes A_+ K^{-1} P \otimes A_- A_+ P \otimes 1 \\&+ \frac{q^{-1/2}}{q-q^{-1}} K^{-2} P \otimes 1 \otimes K^{-2} P \otimes 1 + K^{-2} P \otimes 1 \otimes A_- A_+ P \otimes 1 \\&- (q-q^{-1}) A_- K^{-1} P \otimes A_+^{2} P \otimes A_- K \otimes 1 \\&+ q^{1/2} K^{2} P \otimes A_+ K \otimes A_- K \otimes 1- q^{1/2} K^{-2}P \otimes A_+ K \otimes A_- K \otimes 1.\\ \Gamma _{\{1,4\}}^q =&\frac{q^{-1/2}}{q-q^{-1}} K^{-2} P \otimes 1 \otimes 1 \otimes K^{-2} P - \frac{{q}^{1/2}}{q-q^{-1}} K^{2} P \otimes 1 \otimes 1 \otimes K^{2} P\\&- (q-q^{-1}) A_- A_+ P \otimes A_+ K \otimes K^{2} P \otimes A_- K \\&+ q^{1/2} K^{2} P \otimes A_+ K \otimes K^{2} P \otimes A_- K \\&-q^{1/2} A_- K^{-1} P \otimes K^{-2} P \otimes K^{-2} P \otimes A_+ K \\ {}&- (q-q^{-1}) A_- K^{-1} P \otimes K^{-2} P \otimes A_+ K^{-1}P \otimes A_- A_+ P \\&-q^{1/2} K^{-2} P \otimes A_+ K \otimes K^{2} P \otimes A_- K + A_- A_+ P \otimes 1 \otimes 1 \otimes K^{2} P \\&-(q-q^{-1}) A_- K^{-1} P \otimes A_+ K^{-1} P \otimes 1 \otimes A_- A_+ P\\ {}&- (q-q^{-1}) A_- K^{-1} P \otimes A_+^{2} P \otimes K^{2} P \otimes A_- K\\&- (q-q^{-1}) A_- K^{-1} P \otimes K^{-2} P \otimes A_+^{2} P \otimes A_- K \\&- (q-q^{-1}) A_- A_+ P \otimes 1 \otimes A_+ K \otimes A_- K \\&- q^{-1/2} A_- K^{-1} P \otimes A_+ K^{-1} P \otimes 1 \otimes K^{-2} P+ q^{-1/2} A_- K^{-1} P \\&\otimes A_+ K^{-1} P \otimes 1 \otimes K^{2} P\\ {}&- q^{-1/2} A_- K^{-1} P \otimes K^{-2} P \otimes A_+ K^{-1} P \otimes K^{-2} P\\ {}&+ q^{-1/2}A_+ K^{-1} P \otimes K^{2} P \otimes K^{2} P \otimes A_- K + K^{-2} P \otimes 1 \otimes 1 \otimes A_- A_+ P \\&+ (q-q^{-1})(q^{1/2}-q^{-1/2}) A_- K^{-1} P \otimes A_+ K^{-1} P \otimes A_+ K \otimes A_- K \\&+ q^{-1/2} A_- K^{-1} P \otimes K^{-2} P \otimes A_+ K^{-1} P \otimes K^{2} P \\&+ q^{1/2} K^{2} P \otimes 1 \otimes A_+ K \otimes A_- K - q^{1/2} K^{-2} P \otimes 1 \otimes A_+ K \otimes A_- K. \end{aligned}$$
$$\begin{aligned} \Gamma _{\{1,2,4\}}^q =&-q^{-1/2} A_- K^{-1} P \otimes 1 \otimes A_+ K^{-1} P \otimes K^{-2} P \\&+ q^{1/2}(q-q^{-1}) A_- K^{-1} P \otimes A_+ K \otimes A_+ K \otimes A_- K \\&-q^{1/2} A_- K^{-1} P \otimes A_+ K \otimes 1 \otimes K^{2} P-q^{1/2} K^{-2} P \\&\otimes A_- K^{-1} P \otimes K^{-2} P \otimes A_+ K \\&+ q^{1/2} K^{-2} P \otimes A_- K^{-1} P \otimes A_+ K^{-1} P \otimes K^{2} P\\ {}&-q^{1/2} A_- K^{-1} P \otimes 1 \otimes K^{-2} P \otimes A_+ K \\&- (q-q^{-1}) K^{-2} P \otimes A_- A_+ P \otimes A_+ K \otimes A_- K \\ {}&- (q-q^{-1})A_- K^{-1} P \otimes 1 \otimes A_+ K^{-1} P \otimes A_- A_+ P \\&+ K^{-2} P \otimes K^{-2} P \otimes 1 \otimes A_- A_+ P - \frac{q^{1/2}}{q-q^{-1}} K^{2} P \otimes K^{2} P \otimes 1 \otimes K^{2} P\quad \\&+ q^{-1/2} A_- K^{-1} P \otimes 1 \otimes A_+ K^{-1} P \otimes K^{2} P \\&- (q-q^{-1}) A_- K^{-1} P \otimes 1 \otimes A_+^{2} P \otimes A_- K \\&- (q-q^{-1}) K^{-2} P \otimes A_- K^{-1} P \otimes A_+ K^{-1} P \otimes A_- A_+ P \\&- q^{-1/2} K^{-2} P \otimes A_- K^{-1} P \otimes A_+ K^{-1} P \otimes K^{-2} P\\&+ q^{-1/2} K^{-2} P \otimes A_+ K^{-1} P \otimes K^{2} P \otimes A_- K \\&+ \frac{q^{-1/2}}{q-q^{-1}} K^{-2} P \otimes K^{-2} P \otimes 1 \otimes K^{-2} P \\&-q^{1/2} K^{-2} P \otimes K^{-2} P \otimes A_+ K \otimes A_- K \\&+ q^{1/2} K^{2} P \otimes K^{2} P \otimes A_+ K \otimes A_- K + A_- A_+ P \otimes K^{2} P \otimes 1 \otimes K^{2} P \\&- q^{-1/2}(q-q^{-1}) A_+ K^{-1} P \otimes A_- K \otimes A_+ K \otimes A_- K \\&- (q-q^{-1}) K^{-2} P \otimes A_- K^{-1} P \otimes A_+^{2} P \otimes A_- K \\ {}&- (q-q^{-1}) A_- A_+ P \otimes K^{2} P \otimes A_+ K \otimes A_- K\\&+ K^{-2} P \otimes A_- A_+ P \otimes 1 \otimes K^{2} P + q^{-1/2} A_+ K^{-1} P \otimes 1 \otimes K^{2} P \otimes A_- K \\&+ q^{-1/2}A_+ K^{-1} P \otimes A_- K \otimes 1 \otimes K^{2} P. \end{aligned}$$

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De Bie, H., De Clercq, H. & van de Vijver, W. The Higher Rank q-Deformed Bannai-Ito and Askey-Wilson Algebra. Commun. Math. Phys. 374, 277–316 (2020). https://doi.org/10.1007/s00220-019-03562-w

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