Study of super-algebraic groups using Hopf algebras

2016 Fiscal Year Final Research Report

• Project/Area Number: 26400035
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Algebra
• Research Institution: University of Tsukuba
• Principal Investigator: MASUOKA Akira 筑波大学, 数理物質系, 准教授 (50229366)
• Research Collaborator: SHIBATA Taiki 岡山理科大学, 理学部, 助教
• Project Period (FY): 2014-04-01 – 2017-03-31
• Keywords: ホップ代数 / スーパー代数群 / テンソル圏 / ハリシュ-チャンドラ対

Outline of Final Research Achievements

The notion of algebraic groups is generalized by that of algebraic super-groups. This generalization is important, because every rigid abelian symmetric tensor category over an algebraically closed field of characteristic zero is realized as the category of finite-dimensional representations of some algebraic super-group, as was proved by P. Deligne. The project aims at characteristic-free study of algebraic super-groups using Hopf-algebraic methods and techniques. A category equivalence between algebraic super-groups defined over a commutative ring and Harish-Chandra pairs is proved. The result is applied to re-construct the so-called Chevalley super-groups over the ring of integers. For algebraic super-groups over a field, the properties, such as (i) solvability, (ii) nilpotency and (ii) the Chevalley-type decomposition, are investigated. The results will be hopefully applied to establish a super-analogue of the generalized Picard-Vessiot Theory produced by the PI and K. Amano.

Free Research Field

代数学