论文标题
小组理论观点的Steenrod代数
The Steenrod algebra from the group theoretical viewpoint
论文作者
论文摘要
J.Milnor在论文“ Steenrod代数及其双重”中确定了双重steenrod代数的结构，该结构是有限类型的分级交换型Hopf代数。我们考虑由mod $ p $ steenrod代数的双霍普夫代数代表的仿射组方案$ g_p $。然后，$ g_p $将有限特征$ p $的主要领域分配给了分级的交换代数$ a _*$，以超过$ a _*$的一组同构法，其组结构由正式功率系列的组成给出。本文的目的是通过使用$ g_p（a _*）$的介绍来显示$ g_p $的某些组理论属性。我们给出了$ G_P $的亚组方案的过滤，我们用来估计$ g_p $的有限亚组方案的下部中央系列的长度。我们还提供了一个连续的商映射$ g_p \ xrightArrow {ρ_0} g_p^{\ langle1 \ rangle} \ xrightArrow {ρ_1} g_p^{\ langle2 \ langle2 \ rangle} g_p^{\ langle k \ rangle} \ xrightArrow {ρ_K} g_p^{\ langle k+1 \ rangle} \ xrightArrow {ρ_{ρ_{k+1}} \ cdots $ cdots $ cdots $ cdots $ cdots $ { $ρ_K$是最大的Abelian子组。
In the paper "The Steenrod algebra and its dual", J.Milnor determined the structure of the dual Steenrod algebra which is a graded commutative Hopf algebra of finite type. We consider the affine group scheme $G_p$ represented by the dual Hopf algebra of the mod $p$ Steenrod algebra. Then, $G_p$ assigns a graded commutative algebra $A_*$ over a prime field of finite characteristic $p$ to a set of isomorphisms of the additive formal group law over $A_*$, whose group structure is given by the composition of formal power series. The aim of this paper is to show some group theoretic properties of $G_p$ by making use of this presentation of $G_p(A_*)$. We give a decreasing filtration of subgroup schemes of $G_p$ which we use for estimating the length of the lower central series of finite subgroup schemes of $G_p$. We also give a successive quotient maps $G_p\xrightarrow{ρ_0}G_p^{\langle1\rangle}\xrightarrow{ρ_1}G_p^{\langle2\rangle}\xrightarrow{ρ_2}\cdots\xrightarrow{ρ_{k-1}} G_p^{\langle k\rangle}\xrightarrow{ρ_k}G_p^{\langle k+1\rangle}\xrightarrow{ρ_{k+1}}\cdots$ of affine group schemes over a prime field ${\boldsymbol F}_p$ such that the kernel of $ρ_k$ is a maximal abelian subgroup.
