1999 Fiscal Year Final Research Report Summary
Monodrony representation of elliptic Artin groups and glonatric construction of elliptic Lie algobas
| Project/Area Number |
10640054
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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 |
Geometry
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| Research Institution | Kitami Institute of Technology |
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Principal Investigator |
YAMADA Hiroshi Kitami Institute of Technology, Professor, 工学部, 助教授 (50210472)
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| Co-Investigator(Kenkyū-buntansha) |
SUZUKI Norio Kitami Institute of Technology, Professor, 工学部, 助教授 (80211986)
SLODOWY Peter Kitami Institute of Technology, Professor, 工学部, 助教授 (90312446)
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| Project Period (FY) |
1998 – 1999
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| Keywords | elliptic Artin groups / simply elliptic singularity / elliptic Hecke algebra / loop group / moduli space / elliptic curve / complex reflection group |
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| Research Abstract |
The concepts of elliptic root system, elliptic Dynkin diagram and elliptic Weyl group were introduced by K.Saito to describe the Milnor lattices and the flat structures of semi-universal deformations for simply elliptic singularities. Furthermore, K.Saito and T.Takebayashi studied generators and relations of elliptic Weyl groups in terms of elliptic Dynkin diagrams (This presentation of elliptic Weyl group is a generalization of Coxeter system.). They also proposed the following problems : find generators and relations of"elliptic Lie algebras", "elliptic Hecke algebras"and elliptic Artin groups (the fundamental groups of the complements of the discriminant for simply elliptic singularities) in terms of the elliptic Dynkin diagrams.
In this reseach period, H.Yamada gave an anser to their problem for the case of elliptic Artin groups and elliptic Hecke algebras as an applivation of the twisted Picard-Lefschetz formula due to A.B.Givental. Namely, He described elliptic Artin groups in term of generators associated to the vertices of elliptic Dynkin diagrams that reflect the geometry of vanishing cycles of simply elliptic singularities. As a by-product, he defined elliptic Hecke algebras (which are subalgebras of Cherednik's double affine Hecke algebras) and constructed finite dimensional irreducible representations of them.
P.Slodowy studied the relation of loop groups and simply elliptic singularities and using the moduli theory of principale G-bundles on an elliptic curve, constructed simply elliptic singularities from loop groups with S.Helmke (RIMS.Kyoto unv.) Their theory is a beautifull extension of Grothendieck-Briskorn's for simple singularities and will become an important theory for the conformal field theory in mathematic physics. He also studied the relation of simple singularities and complex reflection groups.
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