A Laplace operator and harmonics on the quantum complex vector space

Abstract

The aim of this article is to study the q-Laplace operator and q-harmonic polynomials on the quantum complex vector space generated by elements zi,wi, i=1,2,…,n, on which the quantum group GLq(n) [or Uq(n)] acts. The q-harmonic polynomials are defined as solutions of the equation Δqp=0, where p is a polynomial in zi,wi, i=1,2,…,n, and the q-Laplace operator Δq is determined in terms of q-derivatives. The q-Laplace operator Δq commutes with the action of GLq(n). The projector Hm,m′:Am,m′→Hm,m′ is constructed, where Am,m′ and Hm,m′ are the spaces of homogeneous (of degree m in zi and of degree m′ in wi) polynomials and homogeneous q-harmonic polynomials, respectively. By using these projectors, a q-analog of the classical zonal spherical and associated spherical harmonics are constructed. They constitute an orthogonal basis of Hm,m′. A q-analog of separation of variables is given. The quantum algebra Uq(gln), acting on Hm,m′, determines an irreducible representation of Uq(gln). This action is explicitly constructed. The results of the article lead to the dual pair (Uq(sl2),Uq(gln)) of quantum algebras.