Modular Leonard triples

Curtin, Brian

doi:10.1016/j.laa.2007.02.024

Cited by 58 publications

(75 citation statements)

References 10 publications

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“…, N and that (K 2 , K 3 ) and (K 1 , K 3 ) form Leonard pairs. In addition, it follows from this observation that in the realization (3.8) of the Bannai-Ito algebra (3.7) obtained from the operators (3.3), (3.4) and (3.6), the set (K 1 , K 2 , K 3 ) constitutes a Leonard Triple, which have studied intensively for the q-Racah scheme in [5,11].…”

Section: Conclusion: the Leonard Triplementioning

confidence: 96%

“…Our approach consists in constructing the Jordan algebra of the intermediary Casimir operators that appear in the coproduct [6] of three sl −1 (2) algebras; this anticommutator algebra coincides with the Bannai-Ito algebra [22], a special case of the Askey-Wilson algebra introduced in [27]. The two Casimir operators are then shown to form a Leonard pair [3,5,11,16,17,18], an observation which allows to recover the recurrence relation of the Bannai-Ito polynomials for the overlap (Racah) coefficients.…”

Section: Introductionmentioning

confidence: 99%

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The Bannai-Ito polynomials as Racah coefficients of the $sl_{-1}(2)$ algebra

Genest

Vinet

Zhedanov

2014

Proc. Amer. Math. Soc.

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Abstract. The Bannai-Ito polynomials are shown to arise as Racah coefficients for sl −1 (2). This Hopf algebra has four generators including an involution and is defined with both commutation and anticommutation relations. It is also equivalent to the parabosonic oscillator algebra. The coproduct is used to show that the Bannai-Ito algebra acts as the hidden symmetry algebra of the Racah problem for sl −1 (2). The Racah coefficients are recovered from a related Leonard pair.

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Section: Conclusion: the Leonard Triplementioning

confidence: 96%

Section: Introductionmentioning

confidence: 99%

The Bannai-Ito polynomials as Racah coefficients of the $sl_{-1}(2)$ algebra

Genest

Vinet

Zhedanov

2014

Proc. Amer. Math. Soc.

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“…By [6, Lemmas 2.4-2.9], the matrices representing U , U * with respect to the basis which realizes the matrices of Lemma 1.7, 1.8, 1.9, 1.10, and 1.11 are as described here. See [6] for a description of this basis.…”

Section: Proof Of Lemmasmentioning

confidence: 99%

“…Motivated by our work on distance-regular graphs which support a spin model [7] and modular Leonard triples [6], we introduce the notion of a spin Leonard pair. Our starting point is the notion of a Leonard pair.…”

Section: Introductionmentioning

confidence: 99%

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Spin Leonard pairs

Curtin

2007

Ramanujan J

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Let K denote a field, and let V denote a vector space over K of finite positive dimension. A pair A, A * of linear operators on V is said to be a Leonard pair on V whenever for each B ∈ {A, A * }, there exists a basis of V with respect to which the matrix representing B is diagonal and the matrix representing the other member of the pair is irreducible tridiagonal. A Leonard pair A, A * on V is said to be a spin Leonard pair whenever there exist invertible linear operators U , U * on V such that U A = AU , U

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

Huang

2015

Commun. Math. Phys.

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Let F denote an algebraically closed field and assume that q ∈ F is a primitive d th root of unity with d = 1, 2, 4. The universal Askey-Wilson algebra △ q is a unital associative F-algebra defined by generators and relations. The generators are A, B, C and the relations assert that each ofWe show that every finite-dimensional irreducible △ q -module is of dimension less than or equal toMoreover we provide an example to show that the bound is tight.

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Modular Leonard triples

Cited by 58 publications

References 10 publications

The Bannai-Ito polynomials as Racah coefficients of the $sl_{-1}(2)$ algebra

The Bannai-Ito polynomials as Racah coefficients of the $sl_{-1}(2)$ algebra

Spin Leonard pairs

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

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