Finite-Dimensional Irreducible Modules of the Universal Askey–Wilson Algebra

Huang, Hau Wen

doi:10.1007/s00220-015-2467-9

Cited by 35 publications

(34 citation statements)

References 42 publications

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“…It is also clear that the duality in this setting is part of an S 3 symmetry acting simultaneously on

A_{0}, A_{1}, A_{2}

and

α_{0}, α_{1}, α_{2}

. The reparameterization of the Askey–Wilson parameters in Huang [, §3.2] seems to behave nicely under this action of S 3 . It is not clear what would be the effect on the basic representation by the action of the full S 3 symmetry group.…”

Section: Duality For the Askey–wilson Polynomials And The Zhedanov Almentioning

confidence: 95%

Dualities in the q‐Askey Scheme and Degenerate DAHA

Koornwinder

Mazzocco

2018

Stud Appl Math

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The Askey–Wilson polynomials are a four‐parameter family of orthogonal symmetric Laurent polynomials Rnfalse[zfalse] that are eigenfunctions of a second‐order q‐difference operator L, and of a second‐order difference operator in the variable n with eigenvalue z+z−1=2x. Then, L and multiplication by z+z−1 generate the Askey–Wilson (Zhedanov) algebra. A nice property of the Askey–Wilson polynomials is that the variables z and n occur in the explicit expression in a similar and to some extent exchangeable way. This property is called duality. It returns in the nonsymmetric case and in the underlying algebraic structures: the Askey–Wilson algebra and the double affine Hecke algebra (DAHA). In this paper, we follow the degeneration of the Askey–Wilson polynomials until two arrows down and in four different situations: for the orthogonal polynomials themselves, for the degenerate Askey–Wilson algebras, for the nonsymmetric polynomials, and for the (degenerate) DAHA and its representations.

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“…It is also clear that the duality in this setting is part of an S 3 symmetry acting simultaneously on

A_{0}, A_{1}, A_{2}

and

α_{0}, α_{1}, α_{2}

Section: Duality For the Askey–wilson Polynomials And The Zhedanov Almentioning

confidence: 95%

Dualities in the q‐Askey Scheme and Degenerate DAHA

Koornwinder

Mazzocco

2018

Stud Appl Math

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“…In addition, it should be possible to study the spectrum of the Heun-Askey-Wilson q-difference operator using an homogeneous Baxter T-Q relation with nonpolynomial solution [LP14]. Also, for q a root of unity and cyclic representations of the Askey-Wilson algebra, the results here presented may be extended in light of [BGV16,H19].…”

Section: Perspectivesmentioning

confidence: 96%

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

Baseilhac

Pimenta

2019

Nuclear Physics B

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An operator of Heun-Askey-Wilson type is diagonalized within the framework of the algebraic Bethe ansatz using the theory of Leonard pairs. For different specializations and the generic case, the corresponding eigenstates are constructed in the form of Bethe states, whose Bethe roots satisfy Bethe ansatz equations of homogeneous or inhomogenous type. For each set of Bethe equations, an alternative presentation is given in terms of 'symmetrized' Bethe roots. Also, two families of on-shell Bethe states are shown to generate two explicit bases on which a Leonard pair acts in a tridiagonal fashion. In a second part, the (in)homogeneous Baxter T-Q relations are derived. Certain realizations of the Heun-Askey-Wilson operator as second q-difference operators are introduced. Acting on the Q-polynomials, they produce the T-Q relations. For a special case, the Q-polynomial is identified with the Askey-Wilson polynomial, which allows one to obtain the solution of the associated Bethe ansatz equations. The analysis presented can be viewed as a toy model for studying integrable models generated from the Askey-Wilson algebra and its generalizations. As examples, the q-analog of the quantum Euler top and various types of three-sites Heisenberg spin chains in a magnetic field with inhomogeneous couplings, three-body and boundary interactions are solved. Numerical examples are given. The results also apply to the time-band limiting problem in signal processing.MSc: 81R50; 81R10; 81U15.

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“…It holds for n = 0 since R 0 (X) = 0 and S 0 (X) = 0. For n ≥ 1 apply induction hypotheses to (14), (15) with X = Θ i (Y ) and simplify them by using the identities for…”

Section: Applying Lemma 33 a Direct Calculation Yields Thatmentioning

confidence: 99%

Center of the universal Askey–Wilson algebra at roots of unity

Huang

2016

Nuclear Physics B

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Inspired by a profound observation on the Racah-Wigner coefficients of Uq(sl2), the Askey-Wilson algebras were introduced in the early 1990s. A universal analog △q of the Askey-Wilson algebras was recently studied. For q not a root of unity, it is known that Z(△q) is isomorphic to the polynomial ring of four variables. A presentation for Z(△q) at q a root of unity is displayed in this paper. As an application, a presentation for the center of the double affine Hecke algebra of type (C ∨ 1 , C1) at roots of unity is obtained.

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Finite-Dimensional Irreducible Modules of the Universal Askey–Wilson Algebra

Cited by 35 publications

References 42 publications

Dualities in the q‐Askey Scheme and Degenerate DAHA

Dualities in the q‐Askey Scheme and Degenerate DAHA

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

Center of the universal Askey–Wilson algebra at roots of unity

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