The $q$-Onsager Algebra and the Universal Askey-Wilson Algebra

Abstract

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 $♮:{\mathcal{O}}_q\to {\Delta }_q$. We apply $♮$ 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.

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• The Universal Askey-Wilson Algebra
• Some algebra related to $P$-and $Q$-polynomial association schemes
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Cited by in corpus (16)

• The Askey-Wilson algebra and its avatars
• The alternating central extension of the $q$-Onsager algebra
• FRT presentation of classical Askey-Wilson algebras
• Higher Rank Relations for the Askey-Wilson and $q$-Bannai-Ito Algebra
• The alternating PBW basis for the positive part of $U_q({\widehat{\mathfrak{sl}}}_2)$
• An action of the free product ${\mathbb{Z}}_2\star {\mathbb{Z}}_2\star {\mathbb{Z}}_2$ on the $q$-Onsager algebra and its current algebra
• A conjecture concerning the $q$-Onsager algebra
• The alternating central extension for the positive part of $U_q({\widehat{\mathfrak{sl}}}_2)$
• The dual pair $(U_q(\mathfrak{su}(1,1)),{\mathfrak{o}}_{q^{1/2}}(2n))$, $q$-oscillators and Askey-Wilson algebras
• Sklyanin-like algebras for ($q$-)linear grids and ($q$-)para-Krawtchouk polynomials
• An Askey-Wilson Algebra of Rank 2
• On the genus two skein algebra
• Defining relations for quantum symmetric pair coideals of Kac-Moody type
• An Infinite-Dimensional ${\square }_q$-Module Obtained from the $q$-Shuffle Algebra for Affine ${\mathfrak{sl}}_2$
• Finite-dimensional irreducible modules of the universal Askey--Wilson algebra at roots of unity
• Double Lowering Operators on Polynomial