AI 中文总结
本文研究了受限量子群$\mathbf{\overline{u}}_q(\mathfrak{sl_2})$的量子双结构，构建了其代数结构并分类了不可约表示。
AI 中文摘要
本文构建了限制量子群$\mathbf{\overline{u}}_q(\mathfrak{sl_2})$的量子对偶$D(\mathbf{\overline{u}}_q(\mathfrak{sl_2}))$。通过生成元和关系描述其代数结构，给出余乘法$\Delta$、余单位$\varepsilon$和对极$S$，最后在$p = 2$时对其所有不可约表示进行分类。
英文摘要
In this paper, we construct the quantum double $D(\mathbf{\overline{u}}_q(\mathfrak{sl_2}))$ of the restricted quantum group $\mathbf{\overline{u}}_q(\mathfrak{sl_2})$. We describe the algebraic structure of $D(\mathbf{\overline{u}}_q(\mathfrak{sl_2}))$ by generators and relations. Moreover, we give the comultiplication $Δ$, the counit $\varepsilon$ and the antipode $S$, respectively. Finally, we classify all irreducible representations of $D(\mathbf{\overline{u}}_q(\mathfrak{sl_2}))$ when $p=2$.
