Study of generalized quantum groups, Coxeter groupoids and related topics

2019 Fiscal Year Final Research Report

• Project/Area Number: 16K05095
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Algebra
• Research Institution: University of Toyama
• Principal Investigator: Yamane Hiroyuki 富山大学, 学術研究部理学系, 教授 (10230517)
• Project Period (FY): 2016-04-01 – 2020-03-31
• Keywords: ホップ代数 / 一般化された量子群 / ワイル亜群 / コクセター亜群 / スーパーリー代数

Outline of Final Research Achievements

As for algebra, which is a branch of mathematics, one of the main topics is studying representation theory of groups. More than one hundred years, specialists have studied representation theory of Lie groups, which can be studied by using calculus. In many cases, it can be reduced to studying Lie algebras, which can be studied by mostly only using linear algebras. The characters of finite dimensional irreducible representations of finite dimensional complex simple Lie algeras can be decided by using the Weyl character formula, which was introduced in 1920's. It can also be proved in an essential way by using the Harish-Chandra isomorphism theorem, which was introduced in 1950's. The main results during the period of this fund are obtaining counterparts of this two theorems for the generalized quantum groups.

Free Research Field

代数学

Academic Significance and Societal Importance of the Research Achievements

関孝和の功績にもあるように，数学において行列式は，常に重要な道具であるとともにそれ自体が研究すべき対象である。行列式は，面積や体積などの計算に用いられるのにもかかわらず負の項が出てくることの意味を洞察することから今では数学の多くの分野で使われるコクセター群の発見に繋がっている。山根宏之はコクセター群のさらなる一般化であるコクセター半群を2008年の共著論文で発見し、その一般化された量子群U(χ）の表現論への応用を追い求めてきた。当該研究期間内にU(χ）の表現論の重大な進展であるHarish-Chandra型の同型定理（共著）とWeyl・Kac型の典型指標公式を得た。