The Bannai-Ito algebra and some applications

Abstract

The Bannai-Ito algebra is presented together with some of its applications. Its relations with the Bannai-Ito polynomials, the Racah problem for the $s{l}_{-1}(2)$ algebra, a superintegrable model with reflections and a Dirac-Dunkl equation on the 2-sphere are surveyed.

arXiv commentProceedings of Group30, Ghent, July 2014. Based on plenary talk given by Luc Vinet at this conference

References in corpus (13)

• A "missing" family of classical orthogonal polynomials
• Classical and Quantum Superintegrability with Applications
• The Dunkl oscillator in the plane II : representations of the symmetry algebra
• The Dunkl oscillator in the plane I : superintegrability, separated wavefunctions and overlap coefficients
• The Dunkl oscillator in three dimensions
• Finite oscillator models: the Hahn oscillator
• The Bannai-Ito algebra and a superintegrable system with reflections on the 2-sphere
• The su(2)_α Hahn oscillator and a discrete Hahn-Fourier transform
• Hopf algebraic structure of the parabosonic and parafermionic algebras and paraparticle generalization of the Jordan Schwinger map
• The algebra of dual -1 Hahn polynomials and the Clebsch-Gordan problem of sl_{-1}(2)
• The Hahn superalgebra and supersymmetric Dunkl oscillator models
• Jordan algebras and orthogonal polynomials
• The anticommutator spin algebra, its representations and quantum group invariance

Cited by in corpus (21)

• Dunkl-Klein-Gordon equation in three-dimensions: The Klein-Gordon oscillator and Coulomb Potential
• Dunkl-Graphene in constant magnetic field
• A Dirac-Dunkl equation on $S^2$ and the Bannai-Ito algebra
• Finite-Dimensional Irreducible Modules of the Racah Algebra at Characteristic Zero
• On the algebra of symmetries of Laplace and Dirac operators
• The Higgs and Hahn algebras from a Howe duality perspective
• Dunkl-Schrodinger Equation in Higher Dimension
• The Racah algebra as a commutant and Howe duality
• Finite-dimensional irreducible modules of the Bannai--Ito algebra at characteristic zero
• Racah problems for the oscillator algebra, the Lie algebra ${\mathfrak{sl}}_n$, and multivariate Krawtchouk polynomials
• The Heun-Racah and Heun-Bannai-Ito algebras
• The quantum superalgebra ${\mathfrak{osp}}_q(1|2)$ and a $q$-generalization of the Bannai-Ito polynomials
• Doubling (Dual) Hahn Polynomials: Classification and Applications
• The dual pair $Pin(2n)\times \mathfrak{osp}(1|2)$, the Dirac equation and the Bannai-Ito algebra
• Dunkl-Klein-Gordon Equation in Higher Dimensions
• The Racah Algebra as a Subalgebra of the Bannai-Ito Algebra
• Finite-dimensional modules of the universal Racah algebra and the universal additive DAHA of type $(C_1^{\vee },C_1)$
• $\mathfrak{osp}(1,2)$ and generalized Bannai-Ito algebras
• Bannai-Ito algebras and the $osp(1,2)$ superalgebra
• Howe duality and algebras of the Askey-Wilson type: an overview
• A Howe correspondence for the algebra of the $\mathfrak{osp}(1|2)$ Clebsch-Gordan coefficients