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A Study on Galois Theory for the Actions of Hopf Algebras

霍普夫代数作用的伽罗瓦理论研究

基本信息

批准号：
10640052
负责人：
YANAI Tadashi
金额：
$ 1.02万
依托单位：
Niihama National College of Technology
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1998
资助国家：
日本
起止时间：
1998 至 1999
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-10640052/
关键词：
Hopf algebra Galois theory ring theory

项目摘要

Let R be a prime ring, Q the symmetric Martindale quotient ring of R and H a finite dimensional pointed Hopf algebra (over a field) with antipode S acting on Q in a continuous and X-outer way. SィイD4-ィエD4 represents the inverse map of S, K the center of Q, and K#H the smash product algebra.Suppose that the following condition is satisfied : (*) any right comodule subalgebra of K#H containing K is a Frobenius extension over K.Then, the following results were obtained.1. For a right H-comodule subalgebra Λ ⊆ K#H containing K, any object in ィイD2ΛィエD2MィイD1HィエD1 is free over Λ.2. The set of left integrals of Λ is a 1-dimensional right K-space and a nonzero left integral generates Λ in ィイD2KィエD2MィイD3H(/)KィエD3.3. Define the maps κ,μ : K#H → K#H by κ(α#h) = ΣSィイD4-ィエD4hィイD21ィエD2 ・ α#ShィイD22ィエD2 and μ(α#h) = ΣSィイD4-ィエD4μηζααhα, where α∈ K, h∈ H. Then, if ζ(resp. η) is a left (resp. right) integral of Λ, there exists α,α' ∈ K and group like elements σ,σ' ∈ H so that κ(ζ) = (α#δ)η and μ(η) = ζ(α'#δ').4. There exists a one to one Galois-type correspondence between the set of all rationally complete subrings R containing the subring of invariants RィイD1HィエD1 and the set of all right comodule subalgebrans of K#H containing K. This result generalized the preceding study concerning the Galois correspondence of the actions of pointed Hopf algebras.We further research whether the condition (*) is satisfied for any continuous and X-outer action of finite dimensional pointed Hopf algebra.

设R是素环，Q是R的对称Martindale商环，H是域上有限维点Hopf代数，对极S连续X-外作用于Q.设S是S的逆映射，K是Q的中心，K#H是Smash积代数，若满足下列条件：（*）K#H的任何包含K的右余代数是K上的Frobenius扩张，则得到如下结果.对于包含K的右H-余代数Λ K#H，在Λ D2Λ D2 M Λ D1 H Λ D1中的任何对象在Λ.2上是自由的。Λ的左积分集是一个1维右K-空间，一个非零的左积分生成Λ在D 2K D 2 M D 3 H（/）K D 3.3中。定义映射κ，μ：K#H → K#H：κ（α#h）= S D 4-D D 4 h D 21 D 2· α#Sh D 22 D 2，μ（α#h）= S D 4-D D 4 μ η αhα，其中α∈ K，h∈ H.然后，如果ζ（resp.左（左），右（右）。右）积分，存在α，α' ∈ K和类群元σ，σ' ∈ H，使得κ（ε）=（α#δ）η和μ（η）= ε（ε '#ε'）.在包含不变量R的子环R的所有有理完备子环R的集合与包含K的K#H的所有右余余代数的集合之间存在一对一的Galois型对应。这一结果推广了前人关于点Hopf代数作用的Galois对应的研究，进一步研究了有限维点Hopf代数的任意连续X-外作用是否满足条件（*）.