Harmonics on the Quantum Euclidean Space Related to the Quantum Orthogonal Group

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dc.creatorIorgov, N. Z.
dc.creatorKlimyk, A. U.
dc.date2003-02-11
dc.date.accessioned2026-07-07T04:55:11Z
dc.date.available2026-07-07T04:55:11Z
dc.descriptionThe aim of this paper is to study harmonic polynomials on the quantum Euclidean space E^N_q generated by elements x_i, i=1,2,...,N, on which the quantum group SO_q(N) acts. The harmonic polynomials are defined as solutions of the equation Δ_q p=0, where p is a polynomial in x_i, i=1,2,...,N, and the q-Laplace operator Δ_q is determined in terms of the differential operators on E^N_q. The projector H_m: {cal A}_m\to {\cal H}_{m} is constructed, where {\cal A}_{m} and {\cal H}_m are the spaces of homogeneous of degree m polynomials and homogeneous harmonic polynomials, respectively. By using these projectors, a q-analogue of the classical zonal polynomials and associated spherical polynomials with respect to the quantum subgroup SO_q(N-2) are constructed. The associated spherical polynomials constitute an orthogonal basis of {\cal H}_m. These polynomials are represented as products of polynomials depending on q-radii and x_j, x_{j'}, j'=N-j+1. This representation is in fact a q-analogue of the classical separation of variables. The dual pair (U_q(sl_2), U_q(so_n)) is related to the action of SO_q(N) on E^N_q. Decomposition into irreducible constituents of the representation of the algebra U_q(sl_2)\times U_q(so_n) defined by the action of this algebra on the space of all polynomials on E^N_q is given.
dc.description17 pages, LaTeX
dc.identifierhttps://arxiv.org/abs/math/0302119
dc.identifierhttp://arxiv.org/abs/math/0302119
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/66492
dc.subjectQuantum Algebra
dc.subjectMathematical Physics
dc.titleHarmonics on the Quantum Euclidean Space Related to the Quantum Orthogonal Group
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