关于某些仿射代数约化群的霍普夫2-上循环
Hopf $2$-cocycles for certain affine algebraic reductive groups
浏览论文内容
中文总结 AI 辅助
本文受仿射代数约化群\(G\)的霍普夫2-上循环分类问题启发，利用相关命题对单位连通分支为环面\(T\)的\(G\)的（极小）霍普夫2-上循环分类，通过分类有限不可分解半单模范畴及秩为1的模范畴，证明纤维函子与霍普夫2-上循环一一对应。
中文摘要 AI 辅助
受复仿射代数约化群\(G\)的霍普夫2-上循环分类这一开放问题的启发，特别是受与极小霍普夫2-上循环相关的文献[EG1，问题7.2]的启发，我们对单位连通分支为环面\(T\)的仿射代数约化群\(G\)的（极小）霍普夫2-上循环进行分类。为此，我们利用文献[ENO，命题5.4]对无限半单等变化张量范畴\(\Rep(T)^K\simeq \Rep(G)\)（其中\(K := G/T\)）上的有限不可分解半单模范畴进行分类。然后用它对\(\Rep(G)\)上秩为1的模范畴进行分类，并表明\(\Rep(G)\)上相应的纤维函子是经典的（即保持维数），这意味着它们与\(G\)的霍普夫2-上循环一一对应。
英文摘要
Let $G$ be an affine algebraic reductive group over $\mathbb{C}$ whose identity component is a torus $T$, and let $K:=G/T$. We realize $\Rep(G)$ as an equivariantization $\Rep(T)^K$ and describe its finite indecomposable semisimple module categories in terms of equivariant module-category data over $\Rep(T)$. For such an equivariantization, rank one is characterized by transitivity of the induced action on the simple objects of the underlying $\Rep(T)$-module category, and nondegeneracy of a projective cocycle on a point stabilizer. We use this criterion to parametrize fiber functors on $\Rep(G)$, prove that every such fiber functor is classical, hence arises from a Hopf $2$-cocycle on $\mathscr{O}(G)$, and classify (minimal) Hopf $2$-cocycle on $\mathscr{O}(G)$. For commutative direct products $G=T\times K$, we also give the canonical Künneth decomposition of the group of gauge classes, including the mixed component, and compare it with our classification.
