The q-Laplace operator and q-harmonic polynomials on the quantum vector space
Copyright policy )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 x1,x2,…,xn. They are defined as solutions of the equation Δqp=0, where p is a polynomial in x1,x2,…,xn and the q-Laplace operator Δq is determined in terms of q-derivatives. The projector Hm:Am→Hm is constructed, where Am and Hm 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 Hm. A q-analog of separation of variables is given. Representations of the nonstandard q-deformed algebra Uq′(son) [which plays the role of the rotation group SO(n) in the case of classical harmonic polynomials] on the spaces Hm are explicitly constructed.
\(q\)-derivatives, Basic hypergeometric functions, \(q\)-deformed algebra, separation of variables, quantum vector space, \(q\)-Laplace operator, spherical harmonics, Quantum groups (quantized enveloping algebras) and related deformations, Quantum groups and related algebraic methods applied to problems in quantum theory, \(q\)-harmonic polynomials
\(q\)-derivatives, Basic hypergeometric functions, \(q\)-deformed algebra, separation of variables, quantum vector space, \(q\)-Laplace operator, spherical harmonics, Quantum groups (quantized enveloping algebras) and related deformations, Quantum groups and related algebraic methods applied to problems in quantum theory, \(q\)-harmonic polynomials
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