For quantized universal enveloping algebras we construct weight modules by inducing representations of the centralizer of the Cartan subalgebra in the quantized universal enveloping algebra. The induced modules arising from finite-dimensional weight modules the centralizer algebra are studied. In particular, we study the induction of one-dimensional modules, and this is related to the study of commutative subalgebras of the centralizer algebra. For the special case of $U_q(\mathfrak{sl}(2,\mathbb{C}))$ we show that we get the admissible unitary representations corresponding to the non-compact real form $U_q(\mathfrak{su}(1,1))$.

中文翻译：

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量化通用包络代数的非极值权重模块

对于量化的通用包络代数，我们通过在量化的通用包络代数中引入 Cartan 子代数的中心子的表示来构造权重模块。研究了由有限维权模和扶正器代数产生的诱导模。特别是我们研究一维模的归纳，这与中心代数的可交换子代数的研究有关。对于 $U_q(\mathfrak{sl}(2,\mathbb{C}))$ 的特殊情况，我们证明我们得到了对应于非紧实实型 $U_q(\mathfrak{su}( 1,1))$。