2001 Fiscal Year Final Research Report Summary
Representation theory of the quantized enveloping algebras and the quantized enveloping superalgebras
| Project/Area Number |
10640022
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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 | Osaka University |
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Principal Investigator |
YAMANE Hiroyuki Osaka University, Graduate School of Science, Lecturer, 大学院・理学研究科, 講師 (10230517)
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| Co-Investigator(Kenkyū-buntansha) |
MURAKAMI Jun Waseda University, School of Science and Engineering, Professor, 理工学部, 教授 (90157751)
DATE Etsuro Osaka University, Graduate School of Science, Professor, 大学院・理学研究科, 教授 (00107062)
KAWANAKA Noriaki Osaka University, Graduate School of Science, Professor, 大学院・理学研究科, 教授 (10028219)
NAGATOMO Kiyokazu Osaka University, Graduate School of Science, Associate Professor, 大学院・理学研究科, 助教授 (90172543)
WATANABE Takao Osaka University, Graduate School of Science, Associate Professor, 大学院・理学研究科, 助教授 (30201198)
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| Project Period (FY) |
1998 – 2001
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| Keywords | Superalgebras / Quantum groups / Toroidal superalgebras / Vertex operator algebras / Representaion theory / Number theory |
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| Research Abstract |
Yamane gave a Serre type theorem for the affine Lie superalgebras G, namely he gave a presentation of G by the Chevalley generators and defining relations satisfied by them. He also gave a similar result for the affine quantised superalgebras U_qG. He alos gave a a presentation of U_qG of type A(M|N)^<(1)> by the Drinfeld generators and defining relations satisfied by them, and defining relations satisfied by them. Dnlike the non-super case, the defining relations are very complicate. However, by comparing the defining relations of G with the ones of U_qG, we can find out the coincidense of the dimensions of the weight sapaces of the Verma modules of G with the ones of U_qG. Let R = C[s^<±1>,t^<±1>] be the two variable Laurent polynomials ring. Let D be the universal central extention of sl(2|2). Then dim D/sl(2|2) = 2., and D(R) = D 【cross product】 R 【symmetry】 Ω_R/dR is the universal central extention of sl(2|2) 【cross product】 R. He gave a presentation of D(R) by the finite Chevalle
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y generators and finite definig relations, and also did the same thing for the D type affine Lie superalgebra D^<(1)> = D 【cross product】 C[t^<±1>] + Cc. It is easy to describe the kernel of the natural map D(R)→ sl(2|2)(R) by using the generators. By the fact, we can also give a presentation of sl(2|2)(R) by the finite Chevalley generators and infinite definig relations.
Yamane gave a Serre type theorem for the affine Lie superalgebras G, namely he gave a presentation of G by the Chevalley generators and defining relations satisfied by them. He also gave a similar result for the affine quantised superalgebras U_qG. He alos gave a a presentation of U_qG of type A(M|N)^<(1)> by the Drinfeld generators and defining relations satisfied by them. Unlike the non-super case, the defining relations are very complicate. However, by comparing the defining relations of G with the ones of U_qG, we can find out the coincidence of the dimensions of the weight sapaces of the Verma modules of G with the ones of U_qG. Let R = C[s^<±1>,t^<±1>] be the two variable Laurent polynomial ring. Let D be the universal central extension of sl(2|2). Then dim D|sl(2|2) = 2., and D(R) = D 【cross product】 R 【symmetry】 Ω_R/dR is the universal central extension of sl(2|2) 【cross product】 R. He gave a presentation of D(R) by the finite Chevalley generators and finite defining relations, and also did the same thing for the D type affine Lie superalgebra D^<(1)> = D 【cross product】 C[t^<±1>] + Cc. It is easy to describe the kernel of the natural map D(R) → sl(2|2)(R) by using the generators. By the fact, we can also give a presentation of sl(2|2)(R) by the finite Chevalley generators and infinite defining relations.
Nagatomo has developed the representation theory of vertex operator algebras, and has applied it to problems arising from conformal field theory. One of the important results is the classification of simple modules for the charge conjugation orbifold model, which opened a way to study conformal field theories with central charge more than or equal to one. On the other hand he applied the systematic study for correlation functions to a construction of modular forms and quasi-modular forms, which attracts much attention of those who work on the theory of modular forms. Less
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