1993 Fiscal Year Final Research Report Summary
Representations of Groups, Lie Algebras, and Algebras
| Project/Area Number |
04452004
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| Research Category |
Grant-in-Aid for General Scientific Research (B)
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| Allocation Type | Single-year Grants |
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| Research Field |
代数学・幾何学
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| Research Institution | Osaka University |
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Principal Investigator |
KAWANAKA Noriaki OSAKA UNIV., DEPT.MATHEMATICS PROFESSOR, 理学部, 教授 (10028219)
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| Co-Investigator(Kenkyū-buntansha) |
SATAKE Ikuo OSAKA UNIV., DEPT.MATHEMATICS INSTRUCTOR, 理学部, 助手 (80243161)
YAMANE Hiroyuki OSAKA UNIV., DEPT.MATHEMATICS INSTRUCTOR, 理学部, 助手 (10230517)
MURAKAMI Jun OSAKA UNIV., DEPT.MATHEMATICS ASSOCIATE PROFESSOR, 理学部, 助教授 (90157751)
NAGATOMO Kiyokazu OSAKA UNIV., DEPT.MATHEMATICS ASSOCIATE PROFESSOR, 理学部, 助教授 (90172543)
MIYANISHI Masayoshi OSAKA UNIV., DEPT.MATHEMATICS PROFESSOR, 理学部, 教授 (80025311)
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| Project Period (FY) |
1992 – 1993
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| Keywords | Ernst equation / Knot invariant / Hecke algebra / Quantum group / Singularity / Super Lie algebra / 特異点 / 超リー代数 |
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| Research Abstract |
Nagatomo constructed infinitely many rational solutions of the Ernst equation in general relativity. Murakami constructed a vertex type state model in Turaev's sense for the multi-variable Alexander polynomial. By using this model, a new set of axioms for the multi-variable Alexander polynomial is obtained. Murakami also determined the structure of the centralizer algebra of the mixed tensor representations of the quantum group U_q(gl(n, c)). This algebra can be considered as a generalization of the lwahori Hecke algebra of type A.Using this algebra, Murakami generalized the Yamada polynomial, which is an invariant of embeddings of a spatial graph in S^3. Murakami also investigated representations of the category of tangles using Kontsevich's interated integral, and succeeded in giving a combinatorial description of Kontsevich's integrals of knots, links and tangles. Satake investigated the simple elliptic singularity of type E6^^^, and constructed explicitly the flat theta invariants in the sense of K.Saito. Yamane constructed a new quasi-triangular Hopf algebra by quantizing the enveloping algebras of simple Lie superalgebras. He defined this Hopt algebra by generators and relations similarly as Drinfeld and Jimbo did for simple Lie algebras. His main contribution is the descovery of new kinds of defining relations which have no connterpart in the case of simple Lie algebras. Even for the enveloping algebras of simple Lie superalgebra, these relations were not known in this precise form. Yamane's results will have applications in mathematical physics and low dimensional topology. For example, he gave a formula for the universal R-matrix of his quasi-triangular Hopt algebra, and using this formula he found a representation theoretic interpretation for the R-matrix of Perk and Schultz.
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