A higher rank Racah algebra and the ${\mathbb{Z}}_2^n$ Laplace-Dunkl operator

Abstract

A higher rank generalization of the (rank one) Racah algebra is obtained as the symmetry algebra of the Laplace-Dunkl operator associated to the ${\mathbb{Z}}_2^n$ root system. This algebra is also the invariance algebra of the generic superintegrable model on the $n$-sphere. Bases of Dunkl harmonics are constructed explicitly using a Cauchy-Kovalevskaia theorem. These bases consist of joint eigenfunctions of maximal Abelian subalgebras of the higher rank Racah algebra. A method to obtain expressions for both the connection coefficients between these bases and the action of the symmetries on these bases is presented.

arXiv comment20 pages, various small changes, accepted in J. Phys. A

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• Symmetry algebra for the generic superintegrable system on the sphere
• Racah algebras, the centralizer $Z_n({\mathfrak{sl}}_2)$ and its Hilbert-Poincaré series
• Finite-Dimensional Irreducible Modules of the Racah Algebra at Characteristic Zero
• FRT presentation of classical Askey-Wilson algebras
• The Higgs and Hahn algebras from a Howe duality perspective
• A superintegrable model with reflections on $S^{n-1}$ and the higher rank Bannai-Ito algebra
• Algebraic (super-)integrability from commutants of subalgebras in universal enveloping algebras
• Racah Algebra $R(n)$ from Coalgebraic Structures and Chains of $R(3)$ Substructures
• Higher Rank Relations for the Askey-Wilson and $q$-Bannai-Ito Algebra
• The Racah algebra as a commutant and Howe duality
• A higher rank extension of the Askey-Wilson Algebra
• Hahn polynomials on polyhedra and quantum integrability
• Racah problems for the oscillator algebra, the Lie algebra ${\mathfrak{sl}}_n$, and multivariate Krawtchouk polynomials
• The general Racah algebra as the symmetry algebra of generic systems on pseudo--spheres
• Truncation of the reflection algebra and the Hahn algebra
• Temperley-Lieb, Brauer and Racah algebras and other centralizers of su(2)
• Coupling coefficients of $s{u}_q(1,1)$ and multivariate $q$-Racah polynomials
• The $q$-Bannai-Ito algebra and multivariate $(-q)$-Racah and Bannai-Ito polynomials
• The dual pair $Pin(2n)\times \mathfrak{osp}(1|2)$, the Dirac equation and the Bannai-Ito algebra
• The dual pair $(U_q(\mathfrak{su}(1,1)),{\mathfrak{o}}_{q^{1/2}}(2n))$, $q$-oscillators and Askey-Wilson algebras
• The Racah Algebra as a Subalgebra of the Bannai-Ito Algebra
• Bargmann and Barut-Girardello models for the Racah algebra
• Hahn polynomials for hypergeometric distribution
• An infinite family of Dunkl type superintegrable curved Hamiltonians through coalgebra symmetry: Oscillator and Kepler-Coulomb models
• Construction of polynomial algebras from intermediate Casimir invariants of Lie algebras
• Howe duality and algebras of the Askey-Wilson type: an overview
• A Fock Model and the Segal-Bargmann Transform for the Minimal Representation of the Orthosymplectic Lie Superalgebra $\mathfrak{osp}(m,2|2n)$
• Darboux transformations from the Appell-Lauricella operator
• Generalized quadratic commutator algebras of PBW-type
• Matrix elements of $SO(3)$ in $s{l}_3$ representations as bispectral multivariate functions
• A Howe correspondence for the algebra of the $\mathfrak{osp}(1|2)$ Clebsch-Gordan coefficients
• Superspace realizations of the Bannai-Ito algebra