1999 Fiscal Year Final Research Report Summary
A study of quantum groups by using the theory of unbounded operator algebras
| Project/Area Number |
10640222
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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 |
Global analysis
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| Research Institution | Fukuoka Universityq |
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Principal Investigator |
KUROSE Hideki Fukuoka Univ., Fac. Sci., Professor, 理学部, 教授 (00161795)
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| Co-Investigator(Kenkyū-buntansha) |
NAKAGAMI Yoshiomi Yokohama City Univ., Dept. Sci., Prof., 理学部, 教授 (70091246)
ARAI Asao Hokkaido Univ., Fac. Sci., Professor, 理学部, 教授 (80134807)
INOUE Atsushi Fukuoka Univ., Fac. Sci., Professor, 理学部, 教授 (50078557)
OGI Hidekazu Fukuoka Univ., Fac. Sci., Assistant, 理学部, 助手 (30248471)
NAKAZATO Hiroshi Hirosaki Univ., Fac. Sci. & Tech., Prof., 理工学部, 教授 (10188922)
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| Project Period (FY) |
1998 – 1999
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| Keywords | quantum group / Hopf algebra / Hopf *-algebra / locally compact quantum group / quantum double / double group construction / Woronowicz unbound operator algebra / unbound operator algebra |
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| Research Abstract |
For Hopf (*-) algebras A and B, using an exchanging map S : B 【cross product】 A → A 【cross product】 B we defined a twisted tensor product of those. We showed that twisted tensor product is not but the bicrossed product of a matched pair of the Hopf (*-) algebras A and B and that the quantum double is a particular example of twisted tensor products. For any coquasitriangular Hopf (*-) algebra A, which is not necessarily finite dimensional, we defined a (*-) algebra homomorphism from quantum double of A and itself into the tensor product of A and its quantum enveloping algebra and showed that it is an isomorphism if A is factorizable. This is a generalization of a result by S. Majid for finite dimensional Hopf algebras. Passing through the above homomorphism, for any coquasitriangular Hopf (*-) algebra A, we prove that the quantum double of A and itself is paired with the quantum enveloping algebra of A and the opposite Hopf (*-) algebra of A and that the pairing nondegenerate iff A is factorizable. The result implies that the quantum enveloping algebra of the complex quantum group of a compact quantum Lie group A corresponds to the quantum double of the quantum enveloping algebra of A and the opposite Hopf algebra of A. The quantum Lorentz algebra is an example. We discussed about the modular theory of compact quantum groups at the level of Hopf *- algebra and defined the notion of algebraic Woronowicz algebras. Taking the dual we could think of the modular theory for discrete quantum groups as multiplier Hopf algebras. The modular structure is further defined for the quantum groups which are described as the quantum double or the double group of compact or discrete quantum groups. In the regular bounded representations of algebraic Woronowicz algebras, we got the corresponding Woronowicz algebras as those weak closures.
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