A higher rank Racah algebra and the $\mathbb{Z}_2^n$ Laplace–Dunkl operator

Bie, Hendrik De; Genest, Vincent X.; Vijver, Wouter van de; Vinet, Luc

doi:10.1088/1751-8121/aa9756

Cited by 50 publications

(78 citation statements)

References 32 publications

(86 reference statements)

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“…The first consists in increasing the number of tensor product and in considering the N fold tensor product of su (2). In this case the Racah algebra is replaced by the higher rank algebra introduced in [4]. The quotient which gives the centralizer may be associated to a Bratteli diagram with N rows describing the direct sum decomposition of N fold tensor product.…”

Section: Discussionmentioning

confidence: 99%

Temperley–Lieb, Brauer and Racah algebras and other centralizers of 𝔰𝔲(2)

Crampé

d'Andecy²,

Vinet³

2020

Trans. Amer. Math. Soc.

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In the spirit of the Schur-Weyl duality, we study the connections between the Racah algebra and the centralizers of tensor products of three (possibly different) irreducible representations of su(2). As a first step we show that the Racah algebra always surjects onto the centralizer. We then offer a conjecture regarding the description of the kernel of the map, which depends on the irreducible representations. If true, this conjecture would provide a presentation of the centralizer as a quotient of the Racah algebra. We prove this conjecture in several cases. In particular, while doing so, we explicitly obtain the Temperley-Lieb algebra, the Brauer algebra and the one-boundary Temperley-Lieb algebra as quotients of the Racah algebra. 2

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Section: Discussionmentioning

confidence: 99%

Temperley–Lieb, Brauer and Racah algebras and other centralizers of 𝔰𝔲(2)

Crampé

d'Andecy²,

Vinet³

2020

Trans. Amer. Math. Soc.

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“…For example, µ (6) and (16), we have 5} 2 , and of course this applies to any morphism of the form µ A i with a i − a i−1 > 2. Similarly, one finds from (10) and (16) that…”

Section: 2mentioning

confidence: 65%

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

Clercq

2019

SIGMA

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The higher rank Askey-Wilson algebra was recently constructed in the n-fold tensor product of Uq(sl 2 ). In this paper we prove a class of identities inside this algebra, which generalize the defining relations of the rank one Askey-Wilson algebra. We extend the known construction algorithm by several equivalent methods, using a novel coaction. These allow to simplify calculations significantly. At the same time, this provides a proof of the corresponding relations for the higher rank q-Bannai-Ito algebra. µ {2,4,5,8} 2 µ {2,4,5,8} 3 (Λ) ⊗ 1, with µ {2,4,5,8} 3Again, the idea is that each element of A but the maximum a m corresponds to an application of ∆, whereas τ L fills up the holes. However, as opposed to Definition 2.3, we now run through the elements of A in decreasing order, from right to left. This is why we refer to the algorithm of Definition 2.5 as the left extension process.

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“…The system with Hamiltonian H in (3.7) has been extensively studied in the literature as an important example of a second-order superintegrable system, possessing the maximal possible number of algebraically independent second-order integrals of motion. It is usually referred to as the generic quantum superintegrable system on the sphere, and has attracted a lot of attention recently in connection to multivariate extensions of the Askey scheme of hypergeometric orthogonal polynomials and their bispectral properties, the Racah problem for su (1, 1), representations of the Kohno-Drinfeld algebra, the Laplace-Dunkl operator associated with Z d+1 2 root system; see for instance [8,16,19] and the references therein. The space V d n introduced in Remark 3.3 appears naturally in the analysis as an irreducible module over the associative algebra generated by the integrals of motion, see [17].…”

Section: Multivariable Operators and Polynomialsmentioning

confidence: 99%

Darboux transformations from the Appell-Lauricella operator

Delgado

Fernández

Илиев

2020

Journal of Mathematical Analysis and Applications

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We define two isomorphic algebras of differential operators: the first algebra consists of ordinary differential operators and contains the hypergeometric differential operator, while the second one consists of partial differential operators in d variables and contains the Appell-Lauricella partial differential operator. Using this isomorphism, we construct partial differential operators which are Darboux transformations from polynomials of the Appell-Lauricella operator. We show that these operators can be embedded into commutative algebras of partial differential operators, containing d mutually commuting and algebraically independent partial differential operators, which can be considered as quantum completely integrable systems. Moreover, these algebras can be simultaneously diagonalized on the space of polynomials leading to extensions of the Jacobi polynomials orthogonal with respect to the Dirichlet distribution on the simplex.

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A higher rank Racah algebra and the $\mathbb{Z}_2^n$ Laplace–Dunkl operator

Cited by 50 publications

References 32 publications

Temperley–Lieb, Brauer and Racah algebras and other centralizers of 𝔰𝔲(2)

Temperley–Lieb, Brauer and Racah algebras and other centralizers of 𝔰𝔲(2)

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

Darboux transformations from the Appell-Lauricella operator

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