2003/02/11 by N. Z. Iorgov, Iorgov, N. Z., A. U. Klimyk +1
Mathematics · Physics and Astronomy · #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Quantum Algebra (math.QA) #math-ph #math.MP #math.QA
paper · pdf · doi:10.48550/arxiv.math/0302119
17 pages, LaTeX
arxiv created 2003/02/11 · openalex publication_date 2003/02/11 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The aim of this paper is to study harmonic polynomials on the quantum Euclidean space ENq generated by elements xi, i=1,2,...,N, on which the quantum group SOq(N) acts. The harmonic polynomials are defined as solutions of the equation Δq p=0, where p is a polynomial in xi, i=1,2,...,N, and the q-Laplace operator Δq is determined in terms of the differential operators on ENq. The projector Hm: cal Am→ \cal Hm is constructed, where \cal Am and \cal Hm 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 SOq(N-2) are constructed. The associated spherical polynomials constitute an orthogonal basis of \cal Hm. These polynomials are represented as products of polynomials depending on q-radii and xj, xj', j'=N-j+1. This representation is in fact a q-analogue of the classical separation of variables. The dual pair (Uq(sl2), Uq(son)) is related to the action of SOq(N) on ENq. Decomposition into irreducible constituents of the representation of the algebra Uq(sl2)× Uq(son) defined by the action of this algebra on the space of all polynomials on ENq is given.