| Project/Area Number |
17540055
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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, Associated Professor, 数理科, 助教授 (50220174)
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| Project Period (FY) |
2005 – 2006
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| Project Status |
Completed (Fiscal Year 2006)
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| Budget Amount *help |
¥800,000 (Direct Cost: ¥800,000)
Fiscal Year 2006: ¥500,000 (Direct Cost: ¥500,000)
Fiscal Year 2005: ¥300,000 (Direct Cost: ¥300,000)
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| Keywords | X-K Hopf algebras / Hopf algebras / Galois correspondence / left integrals / subalgebra of invariants |
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| Research Abstract |
Let K be a field and A a X-K Hopf algebra. Assume that A is finite-dimensional as a left K-space and pointed as a K-coalgebra. Then the dimension of A as a right K-space coincides with the dimension as a left K-space.
Let R be a prime algebra having K as the center of the symmetric quotient algebra Q. Suppose that A acts on Q with continuous and outer action. If A has a nonzero left integral, then a rationally complete subalgebra of R containing the subalgebra of H-invariants of R corresponds to a right coideal subalgebra of A by a certain correspondence map. If R is stable under the action of any grouplike element of A, this correspondence map is injective.
Let A^* be the set of all right K-maps from A to K and B a right coideal subalgebra of A. If B has a nonzero left integral t and t generates B as a left A*-module, then the above correspondence map is surjective.
Next let R be a prime algebra over a field k and H a finite-dimensional pointed Hopf algebra acting on R with an outer action. The following two problems, which extend the results when H is a group algebra, were examined in the case of a certain example and positive answers were obtained.
1. Let U be a subalgebra of R containing the subalgebra of H-invariants RH. Does an automorphism of U which fixes RH extend to an automorphism of R?
2. Assume that the center of the symmetric quotient algebra of R coincides with k. Let I be a normal right coideal subalgebra of H and H' the quotient Hopf algebra of H by I. Then is the action of H' on the subalgebra of I-invariants of R outer?
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