2009 Fiscal Year Final Research Report
Hopf algebras and their applications to quantum groups
| Project/Area Number |
18540008
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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 | University of Tsukuba |
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Principal Investigator |
TAKEUCHI Mitsuhiro University of Tsukuba, 大学院・数理物質科学研究科, 教授 (00015950)
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| Co-Investigator(Kenkyū-buntansha) |
NISHIMURA Hirokazu 筑波大学, 大学院・数理物質科学研究科, 講師 (70135614)
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| Co-Investigator(Renkei-kenkyūsha) |
NAITO Satoshi 筑波大学, 大学院・数理物質科学研究科, 准教授 (60252160)
MASUOKA Akira 筑波大学, 大学院・数理物質科学研究科, 准教授 (50229366)
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| Project Period (FY) |
2006 – 2009
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| Keywords | ホップ代数 / 量子群 / ガロア理論 / FRT構成 / bialgebroid, Picard-Vessiot理論 |
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| Research Abstract |
Hopf algebras and quantum algebra theory are applied in the study of quasi-triangular Hopf algebras, braided categories and knot invariants. In this proposal, we have studied the following topics as well as reviewing the above studies from a new viewpoint.
(1) We have studied H Morita theory, i.e., the theory of Morita equivalences between various Hcomodule algebras with a fixed Hopf algebra H from the viewpoint of quantization of Galois theory, and published a joint paper with Caenepeel, Crivei and Marcus in the Journal of Algebra.
(2) We have studied Picard-Vessiot theory, i.e., Galois theory using algebraic groups (or commutative Hopf algebras) instead of usual groups and its generalization towards Galois theory using quantum groups. As its preliminaries we have studied Hopf algebraic approach to the Picard-Vessiot thery and published a joint paper with Katsutoshi Amano and Akira Masuoka in the Handbook of Algebra.
(3) The principal investigator (Takeuchi) introduced the concept of bialgebroids in 1977. This concept is recently studied very actively by Boem, Brzezinski and so on. We have studied the FRT (Faddeev, Reshetikhin, Takhtajan) construction which plays an mportant role in the quantum group theory in the frame work of bialgebroids and published a joint paper with Youichi Shibukawa, Hokkaido University in the Journal of Algebra.
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