A Dirac–Dunkl Equation on S 2 and the Bannai–Ito Algebra

Bie, Hendrik De; Genest, Vincent X.; Vinet, Luc

doi:10.1007/s00220-016-2648-1

Cited by 41 publications

(100 citation statements)

References 28 publications

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“…Goal and outline. The main goal of the present paper is to extend the results of [2] to arbitrary dimension. As shall be seen this extension is involved, and yet the results are instructive and lend themselves to an elegant presentation.…”

Section: Proposition 2 ([19]mentioning

confidence: 99%

“…In this notation, the "full" n-dimensional Dirac-Dunkl operator D given in (2) can be written as D [n] ; similarly, one has x = x [n] . For ≤ n, we shall also use the notation D [ ] and x [ ] for D {1,..., } and x {1,..., } , respectively.…”

Section: 2mentioning

confidence: 99%

“…In the paper [2] by the authors, it was shown that the Bannai-Ito algebra is the symmetry algebra of the Dirac-Dunkl equation on the 2-sphere associated to the Z 3 2 reflection group. In that context, relevant finite-dimensional representations of (12) were constructed, explicit formulas for the basis vectors of these representations were found, and a connection with the Bannai-Ito polynomials was established.…”

Section: Proposition 2 ([19]mentioning

confidence: 99%

“…Thus, borrowing from the terminology of Lie algebras, one can say that the algebra A n has rank (n − 2). Observe that one can easily define another maximal Abelian subalgebra by applying a permutation of S n on Y n ; this is done by taking πY n = Γ π [2] , Γ π [3] , . .…”

Section: Proposition 2 ([19]mentioning

confidence: 99%

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The Z2n Dirac–Dunkl operator and a higher rank Bannai–Ito algebra

Bie

Genest

Vinet

2016

Advances in Mathematics

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Abstract. The kernel of the Z n 2 Dirac-Dunkl operator is examined. The symmetry algebra An of the associated Dirac-Dunkl equation on S n−1 is determined and is seen to correspond to a higher rank generalization of the Bannai-Ito algebra. A basis for the polynomial null-solutions of the DiracDunkl operator is constructed. The basis elements are joint eigenfunctions of a maximal commutative subalgebra of An and are given explicitly in terms of Jacobi polynomials. The symmetry algebra is shown to act irreducibly on this basis via raising/lowering operators. A scalar realization of An is proposed.

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Section: Proposition 2 ([19]mentioning

confidence: 99%

Section: 2mentioning

confidence: 99%

Section: Proposition 2 ([19]mentioning

confidence: 99%

Section: Proposition 2 ([19]mentioning

confidence: 99%

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The Z2n Dirac–Dunkl operator and a higher rank Bannai–Ito algebra

Bie

Genest

Vinet

2016

Advances in Mathematics

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“…It also appears as the hidden algebra behind the Racah problem for the Lie superalgebra osp(1|2) [8] and as a symmetry algebra for superintegrable systems [1,7].…”

Section: The Bannai-ito Algebra and Its Q-extensionmentioning

confidence: 99%

The Equitable Presentation of $${\mathfrak{osp}_q(1|2)}$$ osp q ( 1 | 2 ) and a q-Analog of the Bannai–Ito Algebra

2015

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The equitable presentation of the quantum superalgebra osp q (1|2), in which all generators appear on an equal footing, is exhibited. It is observed that in their equitable presentations, the quantum algebras osp q (1|2) and sl q (2) are related to one another by the formal transformation q → −q. A q-analog of the Bannai-Ito algebra is shown to arise as the covariance algebra of osp q (1|2). AMS classification numbers: 17B37, 20G42, 81R50 IntroductionThe purpose of this Letter is threefold: to display the equitable, or Z 3 -symmetric, presentation of the quantum superalgebra osp q (1|2), to show that the equitable presentations of osp q (1|2) and sl q (2) are related to one another by the formal transformation q → −q, and to demonstrate that the covariance algebra of osp q (1|2) is a q-analog of the Bannai-Ito algebra.Our considerations take root in the Racah problem for the su(2) algebra, i.e. the coupling of three angular momenta. In this problem, the states are usually described in terms of the quantum numbers j i associated to the individual angular momenta J i with i ∈ {1, 2, 3}, the quantum number j associated to the total angular momentum J = J 1 + J 2 + J 3 , the quantum number M associated to the projection of the total angular momentum J along one axis, and any one of the quantum numbers j 12 , j 23 , j 31 associated to the intermediate angular momenta (23), (31)}. These bases are related via Racah coefficients [3]. The main drawback of such bases is their involved behavior under particle permutations. To circumvent this problem, Chakrabarti [2], Lévy-Leblond and Lévy-Nahas [20] devised an "equitable" coupling scheme and showed that there is a "democratic" basis specified by the quantum numbers j 1 , j 2 , j 3 , j and ζ, where ζ is the eigenvalue of the volume operator ∆ = ( J 1 × J 2 ) · J 3 . The three angular momenta J i enter symmetrically in this scheme and the states of the democratic basis have definite behaviors under particle permutations.The Racah-Wilson algebra is the hidden algebraic structure behind the Racah problems of su(2) and su(1, 1) [11]. The concern for a democratic approach to these Racah problems leads to the equitable presentation of this algebra [9]. In this presentation, the defining relations of the RacahWilson algebra are Z 3 -symmetric and all the generators appear on an equal footing, whence the epithets "equitable" or "democratic". It was recently shown in [5] that the equitable generators of the Racah/Wilson algebra can also be realized as quadratic expressions in the equitable sl(2) generators proposed in [14]. Note that the Racah-Wilson algebra also arises as symmetry algebra for superintegrable systems [16] and encodes the bispectrality of the Racah/Wilson polynomials [6].

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