2004 Fiscal Year Final Research Report Summary
A study on duality of Hopf modules
| Project/Area Number |
15540054
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Algebra
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| Research Institution | Niihama National College of Technology |
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Principal Investigator |
YANAI Tadashi Niihama National College of Technology, Engineering Science, Assistant Professor, 数理科, 助教授 (50220174)
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| Project Period (FY) |
2003 – 2004
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| Keywords | Hopf algebras / Hopf modules / Duality / Integrals / Galois correspondence |
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| Research Abstract |
Assume that H is a Hopf algebra over a field k, A a right H-comodule algebra, D the subalgebra of all coinvariants of A, and M the (A,D)-bimodule of all left D-linear maps from A to D. If H is a pointed Hopf algebra, and A is simple as an (A,H)-Hopf module and finite-dimensional as a D-module, then, by twisting the D-module structure of A and the H-comodule structure of M suitably, A is isomorphic to M as an (A,H)-Hopf module and as an (A,D)-bimodule. This result can be considered as a generalization of duality of finite-dimensional Hopf algebras to Hopf modules. Besides, suppose that a finite-dimensional Hopf algebra H acts on a division algebra D and A is a right H-comodule subalgebra of D#H including D. Then this duality implies some properties of integrals in A. Moreover, let R be a prime algebra, K its extended centreoid, and H a finite-dimensional pointed Hopf algebra acting on R by an X-outer action. Then, from those properties of integrals, we have a one to one Galois-type correspondence in arbitrary characteristic between the set of all rationally complete subalgebras of R including the subalgebra of invariants and the set of all right H-comodule subalgebras of K#H including K. For a further study, we have a problem whether these results can be generalized to Hopf algebroids, which are objects including Hopf algebras. We also have a problem whether it is possible to give a Galois correspondence theorem for Hopf algebra actions under a condition which is weaker than that of the result above.
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Research Products
(3 results)
