交换环上关联代数的群分次分类
A classification of group gradings on incidence algebras over commutative rings
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中文总结 AI 辅助
本文分类了交换环上关联代数的群分次，利用本原正交齐次幂等元刻画分次同构类，并给出有限性结果。
中文摘要 AI 辅助
设$R$为含1的交换环，$P$为局部有限偏序集，$G$为群。我们推导了关联代数$I(P,R)$与群代数$RG$之间$R$-代数同构的充分必要条件。然后，对于不可分解环$R$、有限偏序集$P$和任意群$G$，我们分类了$I(P,R)$的$G$-分次（在同构意义下）。该分类基于一组完整的本原正交齐次幂等元。角代数是$G$的有限阿贝尔子群的分裂群代数，非对角Peirce块是由双陪集稳定子的特征标诱导的双模的无重和。我们证明了分次同构具有刚性形式，且一个分次在同构意义下由幂等元偏序集、角群、原子双模的类型及其乘法的结构常数决定。所出现的数据由多项式条件刻画，且在特征零的代数闭域上，具有给定部分不变量的分次同构类仅有有限多个。一个例子表明结构常数不可省略。我们推广并加强了若干已有结果，同时为某些已知事实提供了替代证明。
英文摘要
Let $R$ be a commutative ring with 1, $P$ a locally finite partially ordered set, and $G$ a group. We derive necessary and sufficient conditions for an $R$-algebra isomorphism between the incidence algebra $I(P,R)$ and the group algebra $RG$. Then, for an indecomposable ring $R$, a finite poset $P$ and an arbitrary group $G$, we classify the $G$-gradings of $I(P,R)$ up to graded isomorphism. The classification rests on a complete set of primitive orthogonal homogeneous idempotents. The corner algebras are split group algebras of finite abelian subgroups of $G$, and the off-diagonal Peirce blocks are multiplicity-free sums of bimodules induced from characters of double coset stabilizers. Graded isomorphisms are shown to have a rigid form, and a grading is determined up to graded isomorphism by the poset of idempotents, the corner groups, the types of the atomic bimodules and the structure constants of their multiplication. The data which occur are characterized by polynomial conditions, and over an algebraically closed field of characteristic zero only finitely many graded isomorphism classes share given partial invariants. An example shows that the structure constants cannot be omitted. Some previous results are extended and enhanced, while providing alternative proofs for some known facts.
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- UFSCar（圣卡洛斯联邦大学）
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