Harmonics on the quantum Euclidean space

Abstract

We study harmonic polynomials on the quantum Euclidean space ENq generated by quantum coordinates xi, i = 1, 2, ..., N, on which the quantum group SOq(N) acts. They are defined as solutions of the equation Δqp = 0, where Δq is the q-Laplace operator on ENq. We construct a q-analogue of the classical zonal polynomials and associated spherical polynomials with respect to the quantum subgroup SOq(N − 2). The associated spherical polynomials constitute an orthogonal basis of the spaces of homogeneous harmonic polynomials. They 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.