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The $q$-Onsager Algebra and the Universal Askey–Wilson Algebra
Symmetry, integrability and geometry: methods and applications, Tome 14 (2018) Cet article a éte moissonné depuis la source Math-Net.Ru

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Recently Pascal Baseilhac and Stefan Kolb obtained a PBW basis for the $q$-Onsager algebra $\mathcal O_q$. They defined the PBW basis elements recursively, and it is obscure how to express them in closed form. To mitigate the difficulty, we bring in the universal Askey–Wilson algebra $\Delta_q$. There is a natural algebra homomorphism $\natural \colon \mathcal O_q \to \Delta_q$. We apply $\natural $ to the above PBW basis, and express the images in closed form. Our results make heavy use of the Chebyshev polynomials of the second kind.
Keywords: $q$-Onsager algebra; universal Askey–Wilson algebra; Chebyshev polynomial. 
@article{SIGMA_2018_14_a43,
     author = {Paul Terwilliger},
     title = {The $q${-Onsager} {Algebra} and the {Universal} {Askey{\textendash}Wilson} {Algebra}},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2018},
     volume = {14},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2018_14_a43/}
}
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Paul Terwilliger. The $q$-Onsager Algebra and the Universal Askey–Wilson Algebra. Symmetry, integrability and geometry: methods and applications, Tome 14 (2018). http://geodesic.mathdoc.fr/item/SIGMA_2018_14_a43/

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