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.

群、李代数和李超代数等代数结构具有内部对称性，已在生物学、化学和物理学的许多不同物理应用中使用。表示论是一种通过矩阵实现来编码群和代数信息的方法，在过去一百年中在数学中发挥了基础性作用。表示论的现代趋势涉及使用拓扑（即上同调）和代数几何工具通过张量积结构封装代数对象的表示，从而将表示论与环境几何结构连接起来。该项目将利用表示、几何和上同调之间的深层相互作用作为强大的工具来证明这些代数结构的新结果，并回答与这些表示的几何和同调不变量（如复杂性和非典型性）有关的重要问题。 PI 将组织代数会议，重点关注初级数学家的发展，他将通过研讨会、讲习班和暑期学校的讲座传播表示论现代方法的工作知识。该项目探讨了表示论中感兴趣的核心问题还原代数群、李代数和李超代数理论。这项研究包括发现关于正特征的还原代数群简单模的特征和特征零的经典李超代数的特征的新猜想。还将探讨有限 Chevalley 群和循环代数的不可约表示之间的扩展公式的确定。在本次研究中，PI 将利用张量三角几何和三角类别中的其他几何构造来计算上述代数对象表示的上同调不变量（例如上同调群、模簇）。