论文标题
在组代数的曲折中存在不可或缺的HOPF订单
Existence of integral Hopf orders in twists of group algebras
论文作者
论文摘要
我们发现了一个群体理论条件，在该条件下，以Movshev的方式，组代数的扭曲承认了一个不可或缺的HOPF订单。令$ k $为（足够大的）数字字段，带有整数$ r $。让$ g $为有限的集团，每$ m $ $ m $ abelian子组为$ g $的中心类型。考虑$ k \ hspace {-0.8pt} g $的扭曲$ j $，由$ \ widehat {m} $提供的非排级$ 2 $ cocycle提供。 We show that if there is a Lagrangian decomposition $\widehat{M} \simeq L \times \widehat{L}$ such that $L$ is contained in a normal abelian subgroup $N$ of $G$, then the twisted group algebra $(K\hspace{-0.8pt}G)_J$ admits a Hopf order $X$ over $R$. HOPF订单$ x $构建为$ r $ -subModule，由$ k \ hspace {-1.1pt} n $和$ g $的元素生成。确实是$ k \ hspace {-0.8pt} g $的HOPF订单，因此$ j^{\ pm 1} \ in x \ otimes_r x $。此外，我们为此HOPF命令提供了一些标准，以使其成为独特。我们用几个例子来说明这一结构。作为一个应用程序，我们提供了一个简单且半圣复杂的Hopf代数的示例，该代数不接受整体HOPF订单。
We find a group-theoretical condition under which a twist of a group algebra, in Movshev's way, admits an integral Hopf order. Let $K$ be a (large enough) number field with ring of integers $R$. Let $G$ be a finite group and $M$ an abelian subgroup of $G$ of central type. Consider the twist $J$ for $K\hspace{-0.8pt}G$ afforded by a non-degenerate $2$-cocycle on the character group $\widehat{M}$. We show that if there is a Lagrangian decomposition $\widehat{M} \simeq L \times \widehat{L}$ such that $L$ is contained in a normal abelian subgroup $N$ of $G$, then the twisted group algebra $(K\hspace{-0.8pt}G)_J$ admits a Hopf order $X$ over $R$. The Hopf order $X$ is constructed as the $R$-submodule generated by the primitive idempotents of $K\hspace{-1.1pt}N$ and the elements of $G$. It is indeed a Hopf order of $K\hspace{-0.8pt}G$ such that $J^{\pm 1} \in X \otimes_R X$. Furthermore, we give some criteria for this Hopf order to be unique. We illustrate this construction with several families of examples. As an application, we provide a further example of simple and semisimple complex Hopf algebra that does not admit integral Hopf orders.
