素特征域上典型单李代数的循环群分次
Gradings by cyclic groups on classical simple Lie algebras in prime characteristics
AI总结：
该研究分类了素特征域上典型单李代数的有限循环群分次，将其转化为Weyl型群的轨道问题，推广了Kac坐标分类并明确了特征p下的限制条件。
AI中文摘要：
我们对代数闭域上任意特征的典型单李代数𝔤，按同构分类其有限循环群分次。利用自同构群概形Aut𝔤的光滑性，以及ℤₘ分次与态射μₘ→Aut𝔤的对应关系，将该分类转化为特定Weyl型群的轨道问题。更一般地，对与半单代数群G关联的仿射群概形G，以及与G的根基标架自同构群的子群Γ₀关联的常群概形Γ₀，我们考虑态射μₘ→G⋊Γ₀在G⋊Γ₀共轭下的分类。结果表明，特征p下的分类与特征0下的分类一致，仅特征p下仅允许Γ₀中阶与p互素的元素。对𝔤的ℤₘ分次，这将Kac坐标分类推广到任意特征，但需注意仅允许阶与p互素的图自同构，且当p=2或3时，Aut𝔤的类型未必与𝔤的类型相同。
英文摘要：
We classify, up to isomorphism, gradings by finite cyclic groups on classical simple Lie algebras $\mathfrak{g}$ over an algebraically closed field of arbitrary characteristic. Using the smoothness of the automorphism group scheme $\operatorname{\mathbf{Aut}}\mathfrak{g}$ and correspondence between $\mathbb Z_m$-gradings and morphisms $\boldsymbolμ_m\to\operatorname{\mathbf{Aut}}\mathfrak{g}$, we express the classification as an orbit problem for certain Weyl-type groups. More generally, for the affine group scheme $\mathbf G$ associated to a semisimple algebraic group $G$ and the constant group scheme $\mathbf Γ_0$ associated to a subgroup $Γ_0$ of the automorphism group of the based root datum of $G$, we consider the classification of morphisms $\boldsymbolμ_m\to \mathbf G \rtimes \mathbfΓ_0$ up to conjugation by $G \rtimes Γ_0$. We show that the classification in characteristic $p$ is the same as in characteristic $0$ except that, in characteristic $p$, only elements of $Γ_0$ whose order is prime to $p$ can occur. For $\mathbb{Z}_m$-gradings on $\mathfrak{g}$, this extends the classification by Kac coordinates to arbitrary characteristic, with the caveat that only diagram automorphisms of order prime to $p$ are allowed and, if $p=2$ or $3$, the type of $\operatorname{Aut}\mathfrak{g}$ is not always the same as the type of $\mathfrak{g}$.
