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

TL;DR

This paper extends the $q$-deformed Bannai-Ito and Askey-Wilson algebras to higher ranks, constructing explicit realizations and demonstrating their role as symmetry algebras in a new $q$-Dirac-Dunkl model.

Contribution

The paper introduces a higher rank $q$-Bannai-Ito algebra in tensor products of $rak{osp}_q(1|2)$ and provides an explicit realization with applications to symmetry and module decomposition.

Findings

Constructed the rank $n-2$ $q$-Bannai-Ito algebra $rak{A}_n^q$.

Realized the algebra explicitly using $q$-shift operators and reflections.

Identified the algebra as the symmetry of the $q$-Dirac-Dunkl operator and described its modules.

Abstract

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.