与量子正交群相关的量子欧几里得空间上的调和函数
Harmonics on the Quantum Euclidean Space Related to the Quantum Orthogonal Group
摘要：
本文的目的是研究量子欧几里得空间E^N_q上由元素x_i，i= 1，2，.生成的调和多项式，N，量子群SO_q（N）作用于其上。调和多项式被定义为方程Δ q p=0的解，其中p是x_i中的多项式，i= 1，2，.，N上的微分算子，q-Laplace算子Δ q由E^N_q上的微分算子确定.投影仪H_m：构造了{cal A}_PDMATHABMM { \cal H}_{m}，其中{ \cal A}_{m}和{ \cal H}_m分别是m次齐次多项式空间和齐次调和多项式空间.利用这些投影子，构造了经典带式多项式和相应的球面多项式关于量子子群SO_q（N-2）的q-模拟.相关的球面多项式构成{ \cal H}_m的正交基。这些多项式被表示为取决于q半径和x_j的多项式的乘积，x_{j'}，j '= N-j+1。这种表示实际上是经典分离变量的q-模拟。对偶对（U_q（sl_2），U_q（so_n））与SO_q（N）在E^N_q上的作用有关。给出了由代数U_q（sl_2）\times U_q（so_n）在E^N_q上所有多项式空间上的作用所定义的代数U_q（sl_2）的表示的不可约分解.
英文摘要：
The aim of this paper is to study harmonic polynomials on the quantum Euclidean space E^N_q generated by elements x_i, i=1,2,...,N, on which the quantum group SO_q(N) acts. The harmonic polynomials are defined as solutions of the equation Δ_q p=0, where p is a polynomial in x_i, i=1,2,...,N, and the q-Laplace operator Δ_q is determined in terms of the differential operators on E^N_q. The projector H_m: {cal A}_m\to {\cal H}_{m} is constructed, where {\cal A}_{m} and {\cal H}_m 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 SO_q(N-2) are constructed. The associated spherical polynomials constitute an orthogonal basis of {\cal H}_m. These polynomials are represented as products of polynomials depending on q-radii and x_j, x_{j'}, j'=N-j+1. This representation is in fact a q-analogue of the classical separation of variables. The dual pair (U_q(sl_2), U_q(so_n)) is related to the action of SO_q(N) on E^N_q. Decomposition into irreducible constituents of the representation of the algebra U_q(sl_2)\times U_q(so_n) defined by the action of this algebra on the space of all polynomials on E^N_q is given.
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