A Remark on the Dunkl Differential—Difference Operators

Heckman, Gert

doi:10.1007/978-1-4612-0455-8_8

Cited by 125 publications

(164 citation statements)

References 11 publications

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“…It is well known [13][14][15] that there exists an intertwining operator M = M (g) which establishes an isospectrality of H (g) and H (g+1) . This differential operators of order 1 2 n(n−1) has the simple form 34) and using the Weyl antisymmetry of M it is straightforward to verify that (see e.g.…”

Section: Jhep10(2015)191mentioning

confidence: 99%

The tetrahexahedric angular Calogero model

Correa

Lechtenfeld

2015

J. High Energ. Phys.

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Abstract:The spherical reduction of the rational Calogero model (of type A n−1 and after removing the center of mass) is considered as a maximally superintegrable quantum system, which describes a particle on the (n−2)-sphere subject to a very particular potential. We present a detailed analysis of the simplest non-separable case, n=4, whose potential is singular at the edges of a spherical tetrahexahedron. A complete set of independent conserved charges and of Hamiltonian intertwiners is constructed, and their algebra is elucidated. They arise from the ring of polynomials in Dunkl-deformed angular momenta, by classifying the subspaces invariant and antiinvariant under all Weyl reflections, respectively.

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Section: Jhep10(2015)191mentioning

confidence: 99%

The tetrahexahedric angular Calogero model

Correa

Lechtenfeld

2015

J. High Energ. Phys.

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“…[6]). In the general case the proof of integrability was found later by Heckman and Opdam [7,8] who used very different arguments.…”

Section: Introductionmentioning

confidence: 99%

Deformed Quantum Calogero-Moser Problems and Lie Superalgebras

Сергеев

Веселов

2004

Communications in Mathematical Physics

104

286

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Abstract. The deformed quantum Calogero-Moser-Sutherland problems related to the root systems of the contragredient Lie superalgebras are introduced. The construction is based on the notion of the generalized root systems suggested by V. Serganova. For the classical series a recurrent formula for the quantum integrals is found, which implies the integrability of these problems. The corresponding algebras of the quantum integrals are investigated, the explicit formulas for their Poincare series for generic values of the deformation parameter are presented.

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“…and write ∆ k for the Dunkl Laplacian on R N (see [15]). This is a differentialdifference operator, which reduces to the Euclidean Laplacian ∆ when k ≡ 0.…”

Section: Hermite Semigroup and Fourier Transformmentioning

confidence: 99%

Algebraic Analysis of Minimal Representations

Kobayashi¹

2011

Publ. Res. Inst. Math. Sci.

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Small representations of a group yield large symmetries in the representation space. Analysis of minimal representations utilizes large symmetries in their geometric models, and serves as a driving force in creating new interesting problems that interact with other branches of mathematics.This article discusses the following three topics that arise from minimal representations of the indefinite orthogonal group:1. construction of conservative quantities for ultra-hyperbolic equations, 2. quantitative discrete branching laws, 3. deformation of the Fourier transform, with emphasis on the prominent role of Sato's ideas in algebraic analysis.

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A Remark on the Dunkl Differential—Difference Operators

Cited by 125 publications

References 11 publications

The tetrahexahedric angular Calogero model

The tetrahexahedric angular Calogero model

Deformed Quantum Calogero-Moser Problems and Lie Superalgebras

Algebraic Analysis of Minimal Representations

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