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An embedding of the universal Askey-Wilson algebra into $U_q(\mathfrak{sl}_2)\otimes U_q(\mathfrak{sl}_2)\otimes U_q(\mathfrak{sl}_2)$

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arxiv 1611.02130 v2 pith:UW5OFU7L submitted 2016-10-29 math.QA math-phmath.MP

Hau-Wen Huang This is my paper · ORCID
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keywords mathfrakotimesirreduciblemodulesalgebraalgebrasaskey--wilsonfinite-dimensional
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abstract

The Askey--Wilson algebras were used to interpret the algebraic structure hidden in the Racah--Wigner coefficients of the quantum algebra $U_q(\mathfrak{sl}_2)$. In this paper, we display an injection of a universal analog $\triangle_q$ of Askey--Wilson algebras into $U_q(\mathfrak{sl}_2)\otimes U_q(\mathfrak{sl}_2)\otimes U_q(\mathfrak{sl}_2)$ behind the application. Moreover we formulate the decomposition rules for $3$-fold tensor products of irreducible Verma $U_q(\mathfrak{sl}_2)$-modules and of finite-dimensional irreducible $U_q(\mathfrak{sl}_2)$-modules into the direct sums of finite-dimensional irreducible $\triangle_q$-modules.

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Cited by 4 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Revisiting the Askey--Wilson algebra with the universal R-matrix of $U_q(sl(2))$

    math.QA 2019-08 conditional novelty 7.0 of 10

    A new R-matrix formula defines the third Askey-Wilson generator as a conjugate of the Casimir element in U_q(sl(2))^{⊗3}, and the Askey-Wilson relations are derived from it.

  2. Diagonalization of the Heun-Askey-Wilson operator, Leonard pairs and the algebraic Bethe ansatz

    math-ph 2019-09 conditional novelty 6.0 of 10

    Full diagonalization of the Heun-Askey-Wilson operator via algebraic Bethe ansatz and Leonard pairs, including Bethe equations, T-Q relations, and Askey-Wilson polynomial solutions.

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

    math.QA 2019-08 conditional novelty 6.0 of 10

    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.

  4. The dual pair $\big(U_q(\mathfrak{su}(1,1)),\mathfrak{o}_{q^{1/2}}(2n)\big)$, $q$-oscillators and Askey-Wilson algebras

    math-ph 2019-08 conditional novelty 6.0 of 10

    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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