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

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.