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A class of non-weight modules of 𝑈𝑝(𝖘𝖑2) and Clebsch–Gordan type formulas
Forum Mathematicum ( IF 1.0 ) Pub Date : 2021-05-01 , DOI: 10.1515/forum-2020-0345
Yan-an Cai 1 , Hongjia Chen 2 , Xiangqian Guo 3 , Yao Ma 4 , Mianmian Zhu 3
Affiliation  

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:

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,
更新日期:2021-04-29
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