量子欧几里得空间R_q^N上q-微分算子的厄米性及其形式
On the hermiticity of q-differential operators and forms on the quantum Euclidean spaces R_q^N
摘要：
我们证明了非交换流形R_q^N上U_qso（N）-协变微分学的复杂 *-结构特征可归结为涉及U_qso（N）的中心扩张的带状元及其形式平方根v的相似变换。通过引入非常规的“相似变换”，将R_q^N上函数空间和p-形式空间的子空间化为希尔伯特空间。定义相应标量积的积分中的“权重”，即实现v^{ \pm 1}作用的适当正定q-伪微分算子;这用于使偏q-导数反厄米，并且像往常一样，外部余导数等于外部导数的厄米共轭。在选择沿“径向坐标”r的权重m（r）时存在剩余自由度。除非我们选择一个常数m，否则平方可积函数/形式必须满足一个附加条件，即它们在复r平面上的解析延拓只能在某些特殊格的格点上有极点。在由这个条件自然选择的函数中，有q-特殊函数具有“量化的”自由参数。
英文摘要：
We show that the complicated *-structure characterizing for positive q the U_qso(N)-covariant differential calculus on the non-commutative manifold R_q^N boils down to similarity transformations involving the ribbon element of a central extension of U_qso(N) and its formal square root v. Subspaces of the spaces of functions and of p-forms on R_q^N are made into Hilbert spaces by introducing non-conventional ``weights'' in the integrals defining the corresponding scalar products, namely suitable positive-definite q-pseudodifferential operators realizing the action of v^{\pm 1}; this serves to make the partial q-derivatives antihermitean and the exterior coderivative equal to the hermitean conjugate of the exterior derivative, as usual. There is a residual freedom in the choice of the weight m(r) along the `radial coordinate' r. Unless we choose a constant m, then the square-integrables functions/forms must fulfill an additional condition, namely their analytic continuations to the complex r plane can have poles only on the sites of some special lattice. Among the functions naturally selected by this condition there are q-special functions with `quantized' free parameters.
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