The higher rank $q$-deformed Bannai-Ito and Askey-Wilson algebra

Abstract

The $q$-deformed Bannai-Ito algebra was recently constructed in the threefold tensor product of the quantum superalgebra ${\mathfrak{osp}}_q(1|2)$. 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 ${\mathfrak{osp}}_q(1|2)$. 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.

38 pages, minor changes and references added