Project/Area Number  10640022 
Research Category 
GrantinAid for Scientific Research (C).

Section  一般 
Research Field 
Algebra

Research Institution  Osaka University 
Principal Investigator 
YAMANE Hiroyuki Osaka University, Graduate School of Science, Lecturer, 大学院・理学研究科, 講師 (10230517)

CoInvestigator(Kenkyūbuntansha) 
MURAKAMI Jun Waseda University, School of Science and Engineering, Professor, 理工学部, 教授 (90157751)
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)
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)
原 靖浩 大阪大学, 大学院・理学研究科, 助手 (10294141)
和田 健志 大阪大学, 大学院・理学研究科, 助手 (70294139)

Project Fiscal Year 
1998 – 2001

Project Status 
Completed(Fiscal Year 2001)

Budget Amount *help 
¥3,200,000 (Direct Cost : ¥3,200,000)
Fiscal Year 2001 : ¥700,000 (Direct Cost : ¥700,000)
Fiscal Year 2000 : ¥700,000 (Direct Cost : ¥700,000)
Fiscal Year 1999 : ¥700,000 (Direct Cost : ¥700,000)
Fiscal Year 1998 : ¥1,100,000 (Direct Cost : ¥1,100,000)

Keywords  Superalgebras / Quantum groups / Toroidal superalgebras / Vertex operator algebras / Representaion theory / Number theory / 超代数 / 量子群 / トロイダル超代数 / 頂点作用素代数 / 現論 / 整数論 / スーパーリー代数 / ホップ代数 / 表現論 / 数理物理学 / 偏微分方程式 / 量子包絡超代数 / 岩堀ヘッケ代数 
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(MN)^<(1)> by the Drinfeld generators and defining relations satisfied by them, and defining relations satisfied by them. Dnlike the nonsuper 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(22). Then dim D/sl(22) = 2., and D(R) = D 【cross product】 R 【symmetry】 Ω_R/dR is the universal central extention of sl(22) 【cross product】 R. He gave a presentation of D(R) by the finite Chevalle
… More
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(22)(R) by using the generators. By the fact, we can also give a presentation of sl(22)(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(MN)^<(1)> by the Drinfeld generators and defining relations satisfied by them. Unlike the nonsuper 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(22). Then dim Dsl(22) = 2., and D(R) = D 【cross product】 R 【symmetry】 Ω_R/dR is the universal central extension of sl(22) 【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(22)(R) by using the generators. By the fact, we can also give a presentation of sl(22)(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 quasimodular forms, which attracts much attention of those who work on the theory of modular forms. Less
