Module categories over affine group schemes
Module categories over affine group schemes
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仿射群方案上的模块类别
DOI：
10.4171/qt/59
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发表时间：
2012
期刊：
影响因子：
--
通讯作者：
Shlomo Gelaki
中科院分区：
文献类型：
--
作者：
Shlomo Gelaki
设$k$为特征为$p\ge 0$的代数闭场。设$G$为$k$上的仿射群方案。我们根据$G$上的$(H,\psi)-$等变相干束，对$G$上有限维$k-$矢量空间的相干束的刚性张量范畴$\text{Coh}_f(G)$上的不可分解精确模类进行了分类。由此推导出{\em}$\Rep(G)$上不可分解模类的分类。当$G$是有限的，这就产生了有限张量类别{\em}$\Rep(G)$上不可分解的精确模块类别的分类。特别地，我们得到了有限群方案$G$的群代数$k[G]$的扭曲分类。将此应用于$u(\mathfrak {g})$，其中$\mathfrak {g}$是$k$上的一个具有正特征的有限维$p-$李代数，得到（新）具有正特征的有限维非交换和非协交换三角形Hopf代数。介绍和研究了群方案的理论范畴，并对等范畴有限群方案进行了研究。
Let $k$ be an algebraically closed field of characteristic $p\ge 0$. Let $G$ be an affine group scheme over $k$. We classify the indecomposable exact module categories over the rigid tensor category $\text{Coh}_f(G)$ of coherent sheaves of finite dimensional $k-$vector spaces on $G$, in terms of $(H,\psi)-$equivariant coherent sheaves on $G$. We deduce from it the classification of indecomposable {\em geometrical} module categories over $\Rep(G)$. When $G$ is finite, this yields the classification of {\em all} indecomposable exact module categories over the finite tensor category $\Rep(G)$. In particular, we obtain a classification of twists for the group algebra $k[G]$ of a finite group scheme $G$. Applying this to $u(\mathfrak {g})$, where $\mathfrak {g}$ is a finite dimensional $p-$Lie algebra over $k$ with positive characteristic, produces (new) finite dimensional noncommutative and noncocommutative triangular Hopf algebras in positive characteristic. We also introduce and study group scheme theoretical categories, and study isocategorical finite group schemes.