1998 Fiscal Year Final Research Report Summary
Noncommutative Geometry for Quantum Groups
| Project/Area Number |
09640210
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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 |
解析学
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| Research Institution | Yokohama City University |
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Principal Investigator |
NAKAGAMI Yoshiomi Yokohama City Univ., Math.Sci., Prof., 理学部, 教授 (70091246)
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| Co-Investigator(Kenkyū-buntansha) |
ICHIDA Ryousuke Yokohama City Univ., Math.Sci., Prof., 理学部, 教授 (10094294)
SHIRAISHI Takaaki Yokohama City Univ.Math.Sci.Ass.Prof., 理学部, 助教授 (50143160)
ICHIMURA Fumio Yokohama City Univ.Math.Sci.Ass.Prof., 理学部, 助教授 (00203109)
FUJII Kazuyoki Yokohama City Univ., Math.Sci., Ass.Prof., 理学部, 助教授 (00128084)
MORI Toshio Yokohama City Univ., Math.Sci., Prof., 理学部, 教授 (40046008)
ICHIRAKU Shigeo Yokohama City Univ., Math.Sci., Prof. (30046130)
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| Project Period (FY) |
1997 – 1998
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| Keywords | quantum group / von Neumann algebra / Kac algebra / Woronowicz algebra / Hopf algebra / Lorentz group / C^* -algebra |
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| Research Abstract |
In 1997 : Let's consider the quantum Lorentz group as an example of non-compact quantum groups. Since the Lie algebra sl(2, C) is identified with the complexification of the Lie algebra su(2), the quatum Lorentz group SL_q(2, C) is considered to be the quantum double of the quantum group SL_q(2). We apply this idea to obtain a non-compact Woronowicz algebras corresponding to the quantum Lorentz group. Using the analysis on Woronowicz algebra, we obtain fundamental relations which hold for generators of the quantum enveloping algebra U_q(sl(2, C)). We also find the natural pairing between the quantum group and the quantum enveloping algebra separating to each other. The definition of the quantum enveloping algebra is found to be isomorphic to the definition obtained from the regular functions in the dual space of the coordinate ring for the quantum Lorentz group.
In 1998 : Since the Woronowicz algebra is described in the framework of von Neumann algebras, the action in general is not continuous but measurable, we encounter some troubles in some applications such as non-commutative geometry and so forth. For such reasons we are obliged to define it in the framework of C^*-algebras. While we have known the general operator a1gebraic structure for the quantum groups through the study of Woronowicz algebras, we must solve several non trivial technical difficulties when we go into the argument using the C^*-algebras. More than 5 years has passed since we began to consider such problems together with Woronowicz and Masuda. Now, we can succeed to solve all the mathematical difficulties. In the last few years we spend times to simplify or to revise the manuscript repeatedly from several respects.
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Research Products
(12 results)
