In this paper, we construct a class of new modules for the quantum group Uq⁢(s⁢l2)U_{q}(\mathfrak{sl}_{2}) which are free of rank 1 when restricted to C⁢[K±1]\mathbb{C}[K^{\pm 1}]. The irreducibility of these modules and submodule structure for reducible ones are determined. It is proved that any C⁢[K±1]\mathbb{C}[K^{\pm 1}]-free Uq⁢(s⁢l2)U_{q}(\mathfrak{sl}_{2})-module of rank 1 is isomorphic to one of the modules we constructed, and their isomorphism classes are obtained. We also investigate the tensor products of the C⁢[K±1]\mathbb{C}[K^{\pm 1}]-free modules with finite-dimensional simple modules over Uq⁢(s⁢l2)U_{q}(\mathfrak{sl}_{2}), and for the generic cases, we obtain direct sum decomposition formulas for them, which are similar to the well-known Clebsch–Gordan formula for tensor products between finite-dimensional weight modules over Uq⁢(s⁢l2)U_{q}(\mathfrak{sl}_{2}).

中文翻译：

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𝑈𝑝（𝖘𝖑2）和Clebsch–Gordan类型公式的一类非加权模块

在本文中，我们为量子群Uq⁢（s⁢l2）U_ {q}（\ mathfrak {sl} _ {2}）构造了一类新模块，这些模块在限于C⁢[K]时不属于等级1 ±1] \ mathbb {C} [K ^ {\ pm 1}]。确定了这些模块和子模块结构的不可归约性。证明没有任何C⁢[K±1] \ mathbb {C} [K ^ {\ pm 1}]无Uq⁢（s⁢l2）U_ {q}（\ mathfrak {sl} _ {2}）秩为1的模块与我们构建的模块之一同构，并获得其同构类。我们还研究了在Uq⁢（s⁢l2）U_ {q}上具有有限维简单模块的无C⁢[K±1] \ mathbb {C} [K ^ {\ pm 1}]无模量的张量积（\ mathfrak {sl} _ {2}），对于一般情况，我们为其获得直接和分解公式，