Complex reflection groups and simple singularities

2001 Fiscal Year Final Research Report Summary

• Project/Area Number: 12640054
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Geometry
• Research Institution: Kitami Institute of Technology
• Principal Investigator: YAMADA Hiroshi Kitami Institute of Technology, Prof., 工学部, 教授 (50210472)
• Co-Investigator(Kenkyū-buntansha): WATANABE Fumihiko Kitami Institute of Technology, Ass. Prof., 工学部, 助教授 (20274433) ; SUZUKI Norio Kitami Institute of Technology, Ass. Prof., 工学部, 助教授 (80211986) ; KOUNO Masaharu Kitami Institute of Technology, Prof., 工学部, 教授 (40170203)
• Project Period (FY): 2000 – 2001
• Keywords: elliptic Lie algebra / elliptic Weyl group / SL(2,Z) / elliptic curve / Painleve equation / generalized DS diagram / 1-labeled graph / 1-ラベル付グラフ

Research Abstract

Yamada-Slodowy define an action of SL_2(Z) and a central extension of a double loop group on an elliptic Lie algebra to construct simply elliptic singularities in the view point of Lie algebras. By these action, we deduce the well known action of SL_2(Z) and Heisenberg group on the affine half space h^^〜_H = h x H x C. This explains a geometric and a Lie algebraic nature of them.
To study classical solutions of PainleveVI equation, Watanabe investigate the stratification of the union of all the hyperplanes coming from reflections in the affine Weyl group of D_4^<(1)>. As an application of these results, he determined the configuration of nordal curves on the defining variety of the sixth Painleve equation for each value of the parameter space C^4.
Kouno introduced some notion of generalized DS diagrams and graphs with 1-label for 3-manifolds. Using these notions, he showed that graphs with 1-label determine 3-manifolds which are given by generalized DS diagrams.