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The $q$-Onsager Algebra and the Universal Askey-Wilson Algebra
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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 $\natural \colon \mathcal O_q \to \Delta_q$. 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.
Forward citations
Cited by 3 Pith papers
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Using the quantum torus to investigate the $q$-Onsager algebra
The Baseilhac-Kolb and Lu-Wang root-vector elements of the q-Onsager algebra are expressed as explicit Laurent monomials in the quantum torus.
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Higher Rank Relations for the Askey-Wilson and $q$-Bannai-Ito Algebra
New commutation and q-commutation relations are proven for generators of the higher rank Askey-Wilson and q-Bannai-Ito algebras, extending the rank-one defining relations.
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The dual pair $\big(U_q(\mathfrak{su}(1,1)),\mathfrak{o}_{q^{1/2}}(2n)\big)$, $q$-oscillators and Askey-Wilson algebras
The Askey-Wilson algebra AW(n) arises not only from tensor products of U_q(su(1,1)), but also as the commutant of n commuting rotations in q-oscillator representations of o_{q^{1/2}}(2n), and the two descriptions are ...
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