A Study on Galois Theory for the Actions of Hopf Algebras
Grant-in-Aid for Scientific Research (C)
|Allocation Type||Single-year Grants|
|Research Institution||Niihama National College of Technology|
YANAI Tadashi Niihama Nat. Col. Tech., Eng. Sci., Ass. Prof., 数理科, 助教授 (50220174)
|Project Period (FY)
1998 – 1999
Completed(Fiscal Year 1999)
|Budget Amount *help
¥1,600,000 (Direct Cost : ¥1,600,000)
Fiscal Year 1999 : ¥700,000 (Direct Cost : ¥700,000)
Fiscal Year 1998 : ¥900,000 (Direct Cost : ¥900,000)
|Keywords||Hopf algebra / Galois theory / ring theory|
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.
Research Output (11results)