A representation theoretic approach to the mapping class group of surfaces

Research Project

Project/Area Number	26870368
Research Category	Grant-in-Aid for Young Scientists (B)
Allocation Type	Multi-year Fund
Research Field	Geometry Algebra
Research Institution	The University of Electro-Communications
Principal Investigator	Enomoto Naoya 電気通信大学, 大学院情報理工学研究科, 准教授 (50565710)
Project Period (FY)	2014-04-01 – 2018-03-31
Project Status	Completed (Fiscal Year 2017)
Budget Amount *help	¥3,900,000 (Direct Cost: ¥3,000,000、Indirect Cost: ¥900,000) Fiscal Year 2017: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000) Fiscal Year 2016: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000) Fiscal Year 2015: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000) Fiscal Year 2014: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Keywords	写像類群 / 表現論 / Johnson準同型 / 超平面配置 / 量子群 / ヘッケ環
Outline of Final Research Achievements	The Johnson hoomorphism for the mapping class group of surfaces give an approximation for the Torelli subgroup of the mapping class group in the Lie algebra of the symplectic derivations. In this study, we evaluate the Johnson image in the Lie algebra of symplectic derivations. First, we give a new class (Enomoto-Satoh obstruction) in the cokernel of the Johnson homomorphism using the structure of the Johnson cokernels for the IA-automorphism group for the free group. Second, we detect some new series ("anti-Morita" obstructions and some "hook type obstruction" )of Sp-irreducible representations in the Johnson cokernels for the mapping class group. Our results suggest that the Johnson image is not so big in the Lie algebra of symplectic derivations.