Monodrony representation of elliptic Artin groups and glonatric construction of elliptic Lie algobas
Project/Area Number  10640054 
Research Category 
GrantinAid for Scientific Research (C)

Allocation Type  Singleyear Grants 
Section  一般 
Research Field 
Geometry

Research Institution  Kitami Institute of Technology 
Principal Investigator 
YAMADA Hiroshi Kitami Institute of Technology, Professor, 工学部, 助教授 (50210472)

CoInvestigator(Kenkyūbuntansha) 
SUZUKI Norio Kitami Institute of Technology, Professor, 工学部, 助教授 (80211986)
SLODONY Peter Kitami Institute of Technology, Professor, 工学部, 助教授 (90312446)

Project Period (FY) 
1998 – 1999

Project Status 
Completed(Fiscal Year 1999)

Budget Amount *help 
¥3,300,000 (Direct Cost : ¥3,300,000)
Fiscal Year 1999 : ¥1,200,000 (Direct Cost : ¥1,200,000)
Fiscal Year 1998 : ¥2,100,000 (Direct Cost : ¥2,100,000)

Keywords  elliptic Artin groups / simply elliptic singularity / elliptic Hecke algebra / loop group / moduli space / elliptic curve / complex reflection group / 楕円型ルート系 / 楕円型アルティン群 / 楕円型リー環 / ループ群 / 単純楕円型特異点 / 楕円型ワイル群 / 楕円型ヘッケ環 / アルティン群 / リー環 / モノドロミー / ワイル群 / ルート系 / ディンキン図形 
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 semiuniversal 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 PicardLefschetz 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 byproduct, 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 Gbundles on an elliptic curve, constructed simply elliptic singularities from loop groups with S.Helmke (RIMS.Kyoto unv.) Their theory is a beautifull extension of GrothendieckBriskorn'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.

Report
(3results)
Research Output
(2results)