A Study on Galois Theory for the Actions of Hopf Algebras

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 y D4- y D4 represents theinverse map of S, K the center of Q，和K#H the smash产品algebra.Suppose that the following condition is satisfied：（*） any right comodule subalgebra of k# H containing K is a Frobenius extension over K. then，1. For a right H-comodule subalgebra Λ⊆K#H containing K，any object in in D2Λ D2M in D1H D1 is free over Λ.2. The set of left integrals of Λ is a1-dimensional right K-space and a nonzero left integral generates Λ in i D2K i D2M i D3H(/)K D3.3.Define the maps κ,μ：k # h→k # h byκα# (h) =σsィイd4-ィエd4hィイd21ィエ以及可转换成d2、α# shィイd22ィエ以及可转换成d2 andμ(α# h) =σsィイd4-ィエ引进μηzααhα,where α∈K, h∈h . Then, if ζ(resp. η) is a left (resp. right) integral of Λ，there existsα,α'∈k and group“elementsσ,σ'∈h so thatκ(z) =(α)#当ηandμ(η)= z(α' #当'). 4。There存在s a to one Galois-type correspondence between the set of all rationally completesubrings R containing the subring of invariants R D1H D1 and the set of all right comodulesubalgebrans of K#H containing K. This result generalized the preceding study concerning the Galoiscorrespondence 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。

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.