AI 中文总结
研究特征为零的代数闭域上余中心分裂阿贝尔霍普夫代数扩张，通过\(V\)上归一化群2 - 上链交叉族描述，给出障碍并转化为双特征标提升问题，应用于置换模和考克斯特模算术约化。
AI 中文摘要
我们研究特征为零的代数闭域上的余中心分裂阿贝尔霍普夫代数扩张。其核为\(k^V\)，商为\(k\Gamma\)，其中\(V\)是有限阿贝尔群且\(\Gamma\)作用于\(V\)。对于固定作用，我们通过\(V\)上归一化群2 - 上链的交叉族来描述这些扩张，模齐次截面的变化。我们给出将上同调类提升到此类上链数据的障碍。利用\(V\)的舒尔乘子，我们将此障碍重写为双特征标提升问题；当\(V\)具有奇数指数时它消失。然后我们将该理论应用于置换模和考克斯特模的算术约化，包括明确的二面体和一阶仿射例子。
英文摘要
We study cocentral algebra-split abelian Hopf algebra extensions over an algebraically closed field of characteristic zero for a fixed action of a group on a finite abelian group. We describe the extension classes in terms of crossed families of normalized 2-cocycles and construct the obstruction to representing crossed families of cohomology classes by cocycles satisfying the crossed identity. A bicharacter obstruction maps to the strict lifting obstruction and may remain nonzero even when its image vanishes. For permutation modules, we obtain an explicit description of the corresponding cocentral extension groups. We also study finite reductions of geometric representations of Coxeter groups and compute the first cohomology of the associated linear coefficient modules for finite dihedral groups and the infinite dihedral group.
Comments27 pages. Revised and expanded version. We clarify the distinction between the strict and bicharacter lifting obstructions, compute the full extension group for permutation modules, and extend the infinite-dihedral cohomology calculation. Several proofs and the exposition have also been improved
