Geometric and algebraic constructions in representation theory

表示论中的几何和代数构造

• 批准号: 288307-2010
• 负责人: Dimitrov, Ivan
• 金额: $ 1.09万
• 依托单位: Queen's University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2013
• 资助国家: 加拿大
• 起止时间: 2013-01-01 至 2014-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=525778
• 关键词: Geometric; algebraic; constructions; representation; theory

In representation theory one realizes algebraic objects as linear transformations of vector spaces. In this way abstract mathematical structures can be studied using a concrete description of their elements as matrices. Conversely, concrete structures arising in geometry and physics can be studied by abstract algebraic and combinatorial methods. Representation theory draws ideas and methods from algebra, geometry, and combinatorics and in return provides applications to each of these areas. The interplay between different areas of mathematics has proved extremely fruitful and aesthetically fulfilling. I have successfully employed constructions coming from geometry, algebra, and combinatorics to classification problems in representation theory. Most of my research is focused on studying the structure and representations of infinite dimensional Lie algebras. In a number of cases a new construction allowed a first glimpse into new phenomena distinguishing the infinite dimensional objects from their finite dimensional counterparts.

在表示理论中，人们将代数对象视为向量空间的线性变换。以这种方式，可以使用对其元素作为矩阵的具体描述来研究抽象的数学结构。相反，可以通过抽象代数和组合方法研究在几何和物理学中产生的混凝土结构。表示理论从代数，几何和组合学中汲取思想和方法，作为回报，为每个领域提供了应用。事实证明，不同数学领域之间的相互作用非常富有成果和美观。我已经成功地采用了来自几何形状，代数和组合的结构，用于代表理论中的分类问题。我的大多数研究都集中在研究无限尺寸谎言代数的结构和表示上。在许多情况下，新的结构可以首先瞥见新现象，从而将无限的尺寸对象与有限的尺寸对应物区分开来。