Representation Theory, Geometry, and Cohomology in Tensor Triangulated Categories

张量三角范畴中的表示论、几何和上同调

基本信息

批准号：
1402271
负责人：
Daniel Nakano
金额：
$ 15.83万
依托单位：
University of Georgia Research Foundation Inc
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2014
资助国家：
美国
起止时间：
2014-09-01 至 2018-08-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1402271&HistoricalAwards=false
关键词：
Representation Theory Geometry Cohomology Tensor

项目摘要

Algebraic structures like groups, Lie algebras, and Lie superalgebras have internal symmetries that have been used in many different physical applications in biology, chemistry, and physics. Representation theory, a method to codify information about groups and algebras through matrix realizations, has played a fundamental role in mathematics over the past one hundred years. Modern trends in representation theory have involved using tools from topology (i.e., cohomology) and algebraic geometry to package the representations of an algebraic object via the tensor product structure to connect the representation theory to ambient geometric structures. This project will use these deep interactions between the representations, the geometry, and the cohomology as a powerful device to prove new results about these algebraic structures and to answer important questions pertaining to geometric and homological invariants (like the complexity and atypicality) for these representations. The PI will organize conferences in algebra with an emphasis toward the development of junior mathematicians, and he will disseminate working knowledge of modern methods in representation theory through lectures at seminars, workshops, and summer schools.This project explores central problems of interest in the representation theory of reductive algebraic groups, Lie algebras, and Lie superalgebras. This study includes discovering new conjectures about the characters of simple modules for reductive algebraic groups in positive characteristic, and classical Lie superalgebras in characteristic zero. The determination of formulas for extensions between irreducible representations for finite Chevalley groups and loop algebras will also be explored. In this investigation, the PI will make use of tensor triangulated geometry and other geometric constructions in triangulated categories to compute cohomological invariants (e.g. cohomology groups, module varieties) of representations for the aforementioned algebraic objects.

诸如组，谎言代数和谎言的代数结构具有内部对称性，这些对称性已用于生物学，化学和物理学中的许多不同物理应用中。代表理论是一种通过矩阵实现编纂有关群体和代数信息的方法，在过去一百年中在数学中发挥了基本作用。代表理论的现代趋势涉及使用拓扑（即协同学）和代数几何形状的工具通过张量产品结构包装代数对象的表示形式，以将表示理论连接到环境几何形状结构。该项目将使用表示形式，几何学和共同体之间的这些深层相互作用作为一种强大的工具，以证明有关这些代数结构的新结果，并回答有关这些代表性的几何和同源不变性（如复杂性和非典型性）的重要问题。 PI将在代数中组织会议，重点是发展初级数学家，他将通过研讨会，研讨会和暑期学校的讲座来传播代表理论中现代方法的工作知识。该项目探讨了在代表性的代表性代理群体中兴趣的核心问题。这项研究包括发现了有关阳性特征还原的简单模块特征的新猜想，而特征为零的经典超级甲壳虫。还将探讨确定有限雪佛利组和环代代数的不可减至表示之间扩展的公式。在这项研究中，PI将利用三角形类别中的张量三角几何形状和其他几何构造来计算上述代数对象的表示共同体不变式（例如，共同体学组，模块品种）。