Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 15 (2019), 099, 32 pages      arXiv:1908.11654

Higher Rank Relations for the Askey-Wilson and $q$-Bannai-Ito Algebra

Hadewijch De Clercq
Department of Electronics and Information Systems, Faculty of Engineering and Architecture, Ghent University, Belgium

Received September 03, 2019, in final form December 13, 2019; Published online December 19, 2019

The higher rank Askey-Wilson algebra was recently constructed in the $n$-fold tensor product of $U_q(\mathfrak{sl}_2)$. In this paper we prove a class of identities inside this algebra, which generalize the defining relations of the rank one Askey-Wilson algebra. We extend the known construction algorithm by several equivalent methods, using a novel coaction. These allow to simplify calculations significantly. At the same time, this provides a proof of the corresponding relations for the higher rank $q$-Bannai-Ito algebra.

Key words: Askey-Wilson algebra; Bannai-Ito algebra.

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