### Symmetry, Integrability and Geometry: Methods and Applications From slq(2) to a Parabosonic Hopf Algebra

, 2011

"... Abstract. A Hopf algebra with four generators among which an involution (reflection) operator, is introduced. The defining relations involve commutators and anticommutators. The discrete series representations are developed. Designated by sl−1(2), this algebra encompasses the Lie superalgebra osp(1| ..."

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Abstract. A Hopf algebra with four generators among which an involution (reflection) operator, is introduced. The defining relations involve commutators and anticommutators. The discrete series representations are developed. Designated by sl−1(2), this algebra encompasses the Lie superalgebra osp(1|2). It is obtained as a q = −1 limit of the slq(2) algebra and seen to be equivalent to the parabosonic oscillator algebra in irreducible representations. It possesses a noncocommutative coproduct. The Clebsch–Gordan coefficients (CGC) of sl−1(2) are obtained and expressed in terms of the dual −1 Hahn polynomials. A generating function for the CGC is derived using a Bargmann realization. Key words: parabosonic algebra; dual Hahn polynomials; Clebsch–Gordan coefficients 2010 Mathematics Subject Classification: 17B37; 17B80; 33C45

### unknown title

, 707

"... Direct and reverse log-Sobolev inequalities in µ-deformed Segal-Bargmann analysis ..."

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Direct and reverse log-Sobolev inequalities in µ-deformed Segal-Bargmann analysis

### THE HARDY SPACE H 1 IN THE RATIONAL DUNKL SETTING

"... Abstract. This paper consists in a first study of the Hardy space H 1 in the rational Dunkl setting. Following Uchiyama’s approach, we characterizee H 1 atomically and by means of the heat maximal operator. We also obtain a Fourier multiplier theorem for H 1. These results are proved here in the one ..."

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Abstract. This paper consists in a first study of the Hardy space H 1 in the rational Dunkl setting. Following Uchiyama’s approach, we characterizee H 1 atomically and by means of the heat maximal operator. We also obtain a Fourier multiplier theorem for H 1. These results are proved here in the one-dimensional case and in the product case. hal-00864457, version 1- 21 Sep 2013 1.

### Embedding Theorems for the Dunkl Harmonic Oscillator on the Line

"... Abstract. Embedding results of Sobolev type are proved for the Dunkl harmonic oscillator on the line. ..."

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Abstract. Embedding results of Sobolev type are proved for the Dunkl harmonic oscillator on the line.

### ANALYSIS ON FLAT SYMMETRIC SPACES

"... Abstract. By taking an appropriate zero-curvature limit, we obtain the spheri-cal functions on flat symmetric spaces G0/K as limits of Harish-Chandra’s spher-ical functions. New and explicit formulas for the spherical functions on G0/K are given. For symmetric spaces with root system of type An, we ..."

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Abstract. By taking an appropriate zero-curvature limit, we obtain the spheri-cal functions on flat symmetric spaces G0/K as limits of Harish-Chandra’s spher-ical functions. New and explicit formulas for the spherical functions on G0/K are given. For symmetric spaces with root system of type An, we find the Taylor expansion of the spherical functions on G0/K in a series of Jack polynomials. Résumé. Dans cet article nous obtenons les fonctions sphériques des espaces symétriques plats G0/K comme une limite des fonctions sphériques de Harish-Chandra. Des formules nouvelles et explicites pour les fonctions sphériques sur G0/K sont données. Dans le cas des espaces symétriques de type An, nous par-venons a ̀ donner le développement en série de Taylor des fonctions sphériques sur G0/K, en termes de polynômes de Jack. 1.

### DUNKL OPERATORS AS COVARIANT DERIVATIVES IN A QUANTUM PRINCIPAL BUNDLE

"... Abstract. A quantum principal bundle is constructed for every Coxeter group acting on a finite-dimensional Euclidean space E, and then a connection is also defined on this bundle. The covariant derivatives associated to this connection are the Dunkl operators, originally introduced as part of a prog ..."

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Abstract. A quantum principal bundle is constructed for every Coxeter group acting on a finite-dimensional Euclidean space E, and then a connection is also defined on this bundle. The covariant derivatives associated to this connection are the Dunkl operators, originally introduced as part of a program to generalize harmonic analysis in Euclidean spaces. This gives us a new, geometric way of viewing the Dunkl operators. In particular, we present a new proof of the commutivity of these operators among themselves as a consequence of a geometric property, namely, that the connection has curvature zero. 1.

### FEFFERMAN-STEIN INEQUALITIES FOR THE Z d 2 DUNKL MAXIMAL OPERATOR

, 2008

"... In this article, we establish the Fefferman-Stein inequalities for the Dunkl maximal operator associated with a finite reflection group generated by the sign changes. Similar results are also given for a large class of operators related to Dunkl’s analysis. ..."

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In this article, we establish the Fefferman-Stein inequalities for the Dunkl maximal operator associated with a finite reflection group generated by the sign changes. Similar results are also given for a large class of operators related to Dunkl’s analysis.

### Liouville Theorem for Dunkl Polyharmonic Functions ⋆

, 2008

"... Original article is available at ..."

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