Hidden Gradings in Representation Theory

表示论中的隐藏等级

• 批准号: 1161094
• 负责人: Alexander Kleshchev
• 金额: $ 55.46万
• 依托单位: University of Oregon Eugene
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-06-15 至 2017-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1161094&HistoricalAwards=false
• 关键词: Hidden; Gradings; Representation; Theory

This project is concerned with some diverse projects in representation theory of Lie algebras, finite groups, and related objects such as Hecke algebras, quantum groups, Khovanov-Lauda-Rouquier (KLR) algebras, Brauer algebras, W-algebras, Yangians, and Lie superalgebras. One of the unifying ideas underpinning the proposal is the study of hidden gradings ubiquitous in many important situations in representation theory, as well as a related idea of categorification of quantum objects. Several of the projects are concerned with the representation theory of KLR algebras, which were introduced to categorify quantum groups. In particular the PIs will initiate a study of homological algebra of KLR algebras of finite type, and study a new phenomenon of imaginary Schur-Weyl duality arising from KLR algebras of affine type. Completion of these projects will lead to substantially better understanding of these algebras both in characteristic zero and in positive characteristic. Other projects are concerned with graded category O for the general linear Lie superalgebra, Deligne's category Rep(GL_delta), graded representations of spin symmetric groups, and the Aschbacher-Scott program of classifying maximal subgroups in finite classical groups. The proposal is expected to have applications to several other areas of mathematics including finite group theory (and its applications), Lie theory, combinatorics, representation theory, knot theory and category theory. Representation theory is a core topic in pure mathematics, with many connections to other areas of mathematics, mathematical physics, computer science, chemistry and even biology. In the last few years the subject has been influenced heavily by ideas from higher category theory, leading to the introduction by Khovanov, Lauda and Rouquier of some remarkable new structures known as KLR algebras. These algebras encode higher symmetries underlying a large part of combinatorial representation theory, including classical objects like symmetric and general linear groups. The goal of the project is to build further the theory of these algebras and apply it to improve our understanding of these classical objects. The basic research in this project has potential future broader impacts in computer science and theoretical physics. More directly this award will have important educational impact through the training of graduate students and the on-going efforts of both PIs in mentoring other young researchers in this area. The award will indirectly support the promotion of knowledge of the methods and results of this beautiful subject area both nationally and internationally, through the active involvement of both PIs as organizers of major conferences and as editors of leading specialist journals.

该项目关注李代数，有限群和相关对象的表示论中的一些不同项目，如Hecke代数，量子群，Khovanov-Lauda-Rouquier（KLR）代数，Brauer代数，W-代数，Yangians和李超代数。支持这个提议的统一思想之一是研究在表示论中许多重要情况下普遍存在的隐藏分级，以及量子对象分类的相关思想。其中几个项目涉及KLR代数的表示论，引入KLR代数是为了对量子群进行分类。特别是，PI将启动有限型KLR代数的同调代数的研究，并研究由仿射型KLR代数产生的虚Schur-Weyl对偶的新现象。这些项目的完成将导致更好地了解这些代数的特征零和积极的特点。其他项目涉及一般线性李超代数的分次范畴O，Deligne范畴Rep（GL_delta），自旋对称群的分次表示，以及分类有限经典群中极大子群的Aschbacher-Scott程序。该建议预计将应用于其他几个数学领域，包括有限群论（及其应用），李群理论，组合数学，表示论，纽结理论和范畴理论。表示论是纯数学的核心课题，与数学、数学物理、计算机科学、化学甚至生物学的其他领域有许多联系。在过去的几年中，这一主题已严重影响的想法，从更高的范畴理论，导致引进霍瓦诺夫，劳达和鲁奎尔的一些显着的新结构称为KLR代数。这些代数编码了组合表示论的大部分基础的更高对称性，包括对称和一般线性群等经典对象。该项目的目标是进一步建立这些代数的理论，并应用它来提高我们对这些经典对象的理解。该项目的基础研究在计算机科学和理论物理领域具有潜在的未来更广泛的影响。更直接地说，该奖项将通过对研究生的培训和两个PI在指导该领域其他年轻研究人员方面的持续努力产生重要的教育影响。该奖项将间接支持在国内和国际上推广这一美丽学科领域的方法和成果的知识，通过两个PI作为主要会议的组织者和领先的专业期刊的编辑的积极参与。