1998 Fiscal Year Final Research Report Summary
Completely integrable systema and representation theory of infinite dimen-sional algebras
| Project/Area Number |
09440014
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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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 | Osaka University |
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Principal Investigator |
DATE Etsuro Osaka Univ.Grad.School of Sci.Professor, 大学院・理学研究科, 教授 (00107062)
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| Co-Investigator(Kenkyū-buntansha) |
MIKI Kei Osaka Univ.Grad.School of Sci.Associate Professor, 大学院・理学研究科, 助教授 (40212229)
NAGATOMO Kiyokazu Osaka Univ.Grad.School of Sci.Associate Professor, 大学院・理学研究科, 助教授 (90172543)
SUZUKI Tkashi Osaka Univ.Grad.School of Sci.Professor, 大学院・理学研究科, 教授 (40114516)
KOTANI Shin'ichi Osaka Univ.Grad.School of Sci.Professor, 大学院・理学研究科, 教授 (10025463)
KAWANAKA Noriaki Osaka Univ.Grad.School of Sci.Professor, 大学院・理学研究科, 教授 (10028219)
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| Project Period (FY) |
1997 – 1998
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| Keywords | Onsager algebra / nilpotent Lie algebras / Schur function / Schrodinger operator / semilinear elliptic equation / vertex operator algebra / L operators |
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| Research Abstract |
In a collaboration with Professor S.S.Roan of Academia Sinica (Taipei), the head investigator of this research project has determined the structure of quotient of the Onsager algebra by ideals of it in the case when the quotients do not have central elements. This almost determines the structure of quotients of the Onsager algebra. In particular the case not determined in the previous researches are fixed by finding a relation with the project of classifying nilpotent Lie algebras by classifying the ideals in the nilpotent part of Kac-Moody Lie algebras. To find such relationship computer algebra system on a workstation was very helpful and indispensable, especially in computing examples. The Onsager algebra can be viewed as a Lie algebra deformation of the nilpotent part of the affine Lie algebra A^<(1)>_1. If we identify A^<(1)>_1 with a central extension of the current algebra, the Onsager algebra is a fixed point set of an involution. By using this presentation we can study the structures of ideals of the Onsager algebra. In order to fix the remining case, we used the principal (twisted) realization of A^<(1)>_1. We think that this kind of principal realization will play some important role in the study of deformation of nilpotent parts of other affine Lie algebras in connection with conformal field theory. We are preparing manuscript on this result. Other investigators in this research project have obtained new results in the study of spectrum of one dimensional Schrodinger operator with a random potential, an identity involving Schur functions, positive solutions of sernilinear elliptic partial differential equations, classification of irreducible representations of vertex operator algebras related with free fermions, relationship of L operators in quantum inverse scattering method and Drinfeld's genarators in affine quantum enveloping algebras and other areas.
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