In the hyperalgebra ${{𝒰}}_r$ of the $r$th Frobenius kernel ${({\mathrm{\text{SL}}}_2)}_r$ of the algebraic group ${\mathrm{\text{SL}}}_2$, we construct a basis of the ${{𝒰}}_r$-module generated by a certain element which was given by the author before. As its applications, we also prove some results on the ${{𝒰}}_r$-modules and the algebra ${{𝒰}}_r$.

中文翻译：

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(SL2)r的超代数的某个模块的基础和一些应用

在超代数中${{𝒰}}_r$的$r$th Frobenius 内核${({\mathrm{\text{SL}}}_2)}_r$代数群的${\mathrm{\text{SL}}}_2$，我们构建了一个基础${{𝒰}}_r$- 由作者之前给出的某个元素生成的模块。作为它的应用，我们也证明了一些结果${{𝒰}}_r$-模块和代数${{𝒰}}_r$.