2006 Fiscal Year Final Research Report Summary
The study of quantum toroidal algebras
| Project/Area Number |
16540192
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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 | Osaka University |
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Principal Investigator |
MIKI Kei Osaka University, Graduate school of information science and technology, associate professor, 大学院情報科学研究科, 助教授 (40212229)
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| Co-Investigator(Kenkyū-buntansha) |
DATE Etsuro Osaka University, Graduate school of information science and technology, professor, 大学院情報科学研究科, 教授 (00107062)
YAMANE Hiroyuki Osaka University, Graduate school of information science and technology, associate professor, 大学院情報科学研究科, 助教授 (10230517)
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| Project Period (FY) |
2004 – 2006
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| Keywords | Quantum group / Toroidal Lie algebra / Difference operator / Virasoro algebra |
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| Research Abstract |
1.Miki investigated the following: (1)Let C_γ be the algebra of Laurent polynomials in x and y satisfying the relation xyγ^2yx. The quantum toroidal algebra of type sl_n is a q deformation of the universal enveloping algebra of the derived algebra of the Lie algebra M_n(C_γ). He considered some quotient algebra arising from this algebra and clarified the relation of it to algebras of symmetric Laurent polynomials and Macdonald difference operators. (2)He introduced a q analogue of the universal enveloping algebra of a central extension of the Lie algebra C_γ. For this q analogue the following studies were done. (i)From the tensor product of N representations in terms of one boson he obtained the free field realization of the q deformed W_N algebra (Virasoro algebra in the case N=2) introduced by Frenkel and Reshetikhin, and Shiraishi and his collaborators. (ii)A representation of this algebra is called quasifinte if each weight space of it is finite dimensional. Quasi irreducible highest weight representations were proved to be characterized by some rational function. (iii)The Yangina limit of this algebra was studied.
2.Date studied the quasi-invarinats and locus configurations of Coxeter groups. He also investigated the differential equations for polynomials which arise in the study of the Bethe ansatz for the Chiral Potts model.
3.Yamane obtained a presentation of elliptic (super) Lie algebras of rank【greater than or equal】2 in terms of a finite number of generators and relations. He also classified finite dimensional irreducible representations of Z/3Z quantum groups, which can be regarded as a Z/3Z graded version of super quantum groups. He further introduced a Coxeter groupoid and proved Matsumoto's theorem in a joint work with Heckenberger.
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