The q-Laplace operator and q-harmonic polynomials on the quantum vector space

The aim of this paper is to study q-harmonic polynomials on the quantum vector space generated by q-commuting elements x(1),x(2),...,x(n). They are defined as solutions of the equation Delta (q)p=0, where p is a polynomial in x(1),x(2),...,x(n) and the q-Laplace operator Delta (q) is determined in terms of q-derivatives. The projector H-m:A(m)-->H-m is constructed, where A(m) and H-m are the spaces of homogeneous (of degree m) polynomials and q-harmonic polynomials, respectively. By using these projectors, a q-analog of classical associated spherical harmonics is constructed. They constitute an orthonormal basis of H-m. A q-analog of separation of variables is given. Representations of the nonstandard q-deformed algebra U-q'(so(n)) [which plays the role of the rotation group SO(n) in the case of classical harmonic polynomials] on the spaces H-m are explicitly constructed. (C) 2001 American Institute of Physics.