论文标题
使用weyl groupoid的unity根部与量子特殊线性超级甲虫相关的HOPF超级甲虫的分类
Classification of Hopf superalgebras associated with quantum special linear superalgebra at roots of unity using Weyl groupoid
论文作者
论文摘要
我们使用超类别方法总结了Weylsoid的定义，以研究Unity根部的量子超级甲壳虫。我们展示了如何在量子超级别上的HOPF超级的结构由量子Weyl族类型确定。 $ \ mathfrak {sl}（m | n）$的weyl glopoid构建为某些超类别。我们证明，在这种情况下，与Dynkin图相关的量子超级骨架是同构成的超级甲虫。显示了这些被认为是HOPF超甲虫的量子超级甲虫如何通过曲折和同构连接。我们使用被认为是Weyl量子组元素的lusztig同构来明确构建这些曲折。我们为每个量子超级甲虫建立一个PBW基础，并研究量子超级甲虫如何与它们的经典限制相连i。 e。躺在superbialgebras。我们找到了用于通用$ r $ - matrices的显式乘法公式，描述它们之间的关系，并为量子superalgebra $ u_q（\ mathfrak {sl} {sl}（m | n））$。
We summarize the definition of the Weyl groupoid using supercategory approach in order to investigate quantum superalgebras at roots of unity. We show how the structure of a Hopf superalgebra on a quantum superalgebra is determined by the quantum Weyl groupoid. The Weyl groupoid of $\mathfrak{sl}(m|n)$ is constructed to this end as some supercategory. We prove that in this case quantum superalgebras associated with Dynkin diagrams are isomorphic as superalgebras. It is shown how these quantum superalgebras considered as Hopf superalgebras are connected via twists and isomorphisms. We explicitly construct these twists using the Lusztig isomorphisms considered as elements of the Weyl quantum groupoid. We build a PBW basis for each quantum superalgebra, and investigate how quantum superalgebras are connected with their classical limits, i. e. Lie superbialgebras. We find explicit multiplicative formulas for universal $R$-matrices, describe relations between them for each realization and classify Hopf superalgebras and triangular structures for the quantum superalgebra $U_q(\mathfrak{sl}(m|n))$.
