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

Crampé, Nicolas; Vijver, Wouter van de; Vinet, Luc

doi:10.1007/s00023-020-00972-8

Cited by 13 publications

(12 citation statements)

References 51 publications

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“…We restrict to representations for the symmetric powers of the fundamental representations, then the eigenfunctions can be described in terms of multivariable Krawtchouk polynomials following Iliev [12] establishing them as overlap coefficients between a natural basis for two different Cartan subalgebras. Similar group theoretic interpretations of these multivariable Krawtchouk polynomials have been established by Crampé et al [5] and Genest et al [8]. We discuss briefly the t-dependence of the corresponding eigenvectors of L(t).…”

Section: Introductionsupporting

confidence: 72%

Orthogonal functions related to Lax pairs in Lie algebras

Groenevelt

Koelink

2021

Ramanujan J

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We study a Lax pair in a 2-parameter Lie algebra in various representations. The overlap coefficients of the eigenfunctions of L and the standard basis are given in terms of orthogonal polynomials and orthogonal functions. Eigenfunctions for the operator L for a Lax pair for $$\mathfrak {sl}(d+1,\mathbb {C})$$ sl ( d + 1 , C ) is studied in certain representations.

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Section: Introductionsupporting

confidence: 72%

Orthogonal functions related to Lax pairs in Lie algebras

Groenevelt

Koelink

2021

Ramanujan J

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“…This result appears to be a specific case of the more general one appearing in [17,46], where the authors deal with a general number k of disjoint subsets {K p } p=1,...,k of [n]. We refer to [15] for the proof of this Lemma (see also [19] for the generalisation to the k-fold tensor product). So, the idea is to make use of the above result paraphrasing it in this classical context.…”

Section: Higher Dimensional Classical Substructures Injective Morphis...mentioning

confidence: 87%

“…. , n − 1 with the help of the following [15,17,19,46]: Lemma 3.1. Let {K 1 , K 2 , K 3 } be a set composed by three disjoint subsets of the set [n] := {1, .…”

Section: Higher Dimensional Classical Substructures Injective Morphis...mentioning

confidence: 99%

“…Remarkably, for the two-dimensional case, it has been proved that all second-order superintegrable systems can be obtained from a unique model known as the generic superintegrable system on the 2-sphere [14]. A higher dimensional extension of this model, the generic superintergrable system on the (n − 1)-sphere, has been recently studied in a series of papers (see [15,16,17,18,19] and references therein) devoted to the so-called generalised Racah algebra R(n), a higher rank generalisation of the rank one Racah algebra R(3). Interestingly, the authors proved that the above algebraic structure is naturally realised as the symmetry algebra of the above-mentioned superintegrable model.…”

Section: Introductionmentioning

confidence: 99%

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Racah Algebra $R(n)$ from Coalgebraic Structures and Chains of $R(3)$ Substructures

Latini,

Marquette,

Zhang

2021

Preprint

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The recent interest in the study of higher-rank polynomial algebras related to n-dimensional classical and quantum superintegrable systems with coalgebra symmetry and their connection with the generalised Racah algebra R(n), a higher-rank generalisation of the rank one Racah algebra R(3), raises the problem of understanding the role played by the n − 2 quadratic subalgebras generated by the left and right Casimir invariants (sometimes referred as universal quadratic substructures) from this new perspective. Such subalgebra structures play a signficant role in the algebraic derivation of spectrum of quantum superintegrable systems. In this work, we tackle this problem and show that the above quadratic subalgebra structures can be understood, at a fixed n > 3, as the images of n − 2 injective morphisms of R(3) into R(n). We show that each of the n − 2 quadratic subalgebras is isomorphic to the rank one Racah algebra R(3). As a byproduct, we also obtain an equivalent presentation for the universal quadratic subtructures generated by the partial Casimir invariants of the coalgebra. The construction, which relies on explicit (symplectic or differential) realisations of the generators, is performed in both the classical and the quantum cases.1 Here, and throughout the paper, we shall assume the classical/quantum integrals to be polynomials in the momenta/finiteorder differential operators, so that we are implicitly defining, following the terminology used in [1], polynomial superintegrability/superintegrability of finite-order. In this perspective, the order as a polynomial in the momenta/as a linear differential operator defines the order of the classical/quantum integral of motion.

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“…Other cases where sl 2 is replaced by other algebras are also known. For example the diagonal centralizer of the oscillator algebra has been studied in [41], the diagonal centralizer of the super Lie algebra osp(1|2) is known to be related to the Bannai-Ito algebra [42] and the centralizer of sl 3 in its twofold tensor product has been introduced in [33]. An important generalization concerns the quantum group U q (sl 2 ).…”

Section: Discussionmentioning

confidence: 99%

Racah algebras, the centralizer $Z_n(\mathfrak{sl}_2)$ and its Hilbert-Poincaré series

Crampe,

Gaboriaud,

d'Andecy

et al. 2021

Preprint

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The higher rank Racah algebra R(n) introduced in [1] is recalled. A quotient of this algebra by central elements, which we call the special Racah algebra sR(n), is then introduced. Using results from classical invariant theory, this sR(n) algebra is shown to be isomorphic to the centralizer Z n (sl 2 ) of the diagonal embedding of U (sl 2 ) in U (sl 2 ) ⊗n . This leads to a first and novel presentation of the centralizer Z n (sl 2 ) in terms of generators and defining relations. An explicit formula of its Hilbert-Poincaré series is also obtained and studied. The extension of the results to the study of the special Askey-Wilson algebra and its higher rank generalizations is discussed.

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

Cited by 13 publications

References 51 publications

Orthogonal functions related to Lax pairs in Lie algebras

Orthogonal functions related to Lax pairs in Lie algebras

Racah Algebra $R(n)$ from Coalgebraic Structures and Chains of $R(3)$ Substructures

Racah algebras, the centralizer $Z_n(\mathfrak{sl}_2)$ and its Hilbert-Poincaré series

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