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Huang, Hau Wen

doi:10.1016/j.nuclphysb.2017.07.007

Cited by 30 publications

(24 citation statements)

References 12 publications

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“…The cases with n = 3 for the Lie algebra su(2) (or su(1, 1)), the quantum algebra U q (sl 2 ) and the Lie superalgebra osp(1|2) have first been examined. They have led respectively to the (universal versions of the) Racah algebra R(3) [28,21,23] the Askey-Wilson algebra AW (3) [29,32] and the Bannai-Ito algebra BI(3) [22]. In this picture, where there is an implicit map from the abstract Racah algebra onto the centralizer of the diagonal action of say, su(2), U q (sl 2 ) or osp(1|2) on their triple product, the images of the three generators of the Racah algebra are expressed in terms of the intermediate Casimir elements.…”

Section: Introductionmentioning

confidence: 99%

Racah Problems for the Oscillator Algebra, the Lie Algebra $$\mathfrak {sl}_n$$, and Multivariate Krawtchouk Polynomials

Crampé

Vijver

Vinet

2020

Ann. Henri Poincaré

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The oscillator Racah algebra Rn(h) is realized by the intermediate Casimir operators arising in the multifold tensor product of the oscillator algebra h. An embedding of the Lie algebra sl n−1 into Rn(h) is presented. It relates the representation theory of the two algebras. We establish the connection between recoupling coefficients for h and matrix elements of slnrepresentations which are both expressed in terms of multivariate Krawtchouk polynomials of Griffiths type.

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Section: Introductionmentioning

confidence: 99%

Racah Problems for the Oscillator Algebra, the Lie Algebra $$\mathfrak {sl}_n$$, and Multivariate Krawtchouk Polynomials

Crampé

Vijver

Vinet

2020

Ann. Henri Poincaré

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“…Example 4. [GZ93b,H16] With respect to Example 2, consider the tensor product of three irreducible repre-…”

Section: Diagonalization Of the Heun-askey-wilson Operator Via The Almentioning

confidence: 99%

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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An embedding of the universal Askey–Wilson algebra intoUq(sl2)⊗U

Cited by 30 publications

References 12 publications

Racah Problems for the Oscillator Algebra, the Lie Algebra $$\mathfrak {sl}_n$$, and Multivariate Krawtchouk Polynomials

Racah Problems for the Oscillator Algebra, the Lie Algebra $$\mathfrak {sl}_n$$, and Multivariate Krawtchouk Polynomials

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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