Cohomological Methods in the Representation Theory of Algebraic Groups, Quantum Groups and Superalgebras

代数群、量子群和超代数表示论中的上同调方法

• 批准号: 0654169
• 负责人: Daniel Nakano
• 金额: $ 15.97万
• 依托单位: University of Georgia Research Foundation Inc
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-09-15 至 2011-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0654169&HistoricalAwards=false
• 关键词: Cohomological; Methods; Representation; Theory; Algebraic

The principal investigator (PI) will investigate problems involving cohomological methods with applications to the representation theory of algebraic groups, quantum groups, and Lie superalgebras. More specifically, the PI aims to expand our knowledge of important cohomological calculations for algebraic groups, quantum groups and Lie algebras. Studying line bundle cohomology for the flag variety will play a prominent role in these calculations through a series of steps involving Lie algebra and Frobenius kernel cohomology. The PI proposes to utilize geometric methods to study the representation theory through the use of support varieties. For Lie superalgebras, this approach provides a beautiful homological interpretation of the well-studied combinatorial notions of defect and atypicality. Representation theory emerged about 100 years ago with the pioneering work of Frobenius and Schur, and has become a central area of mathematics because of its connections to combinatorics, algebraic geometry, number theory, and applications to physics. Cohomology theories were developed throughout the 20th century by topologists to construct algebraic invariants for the investigation of manifolds and topological spaces. Cohomology was also defined for algebraic structures like groups and Lie algebras to determine ways in which their representations can be glued together. Even more striking is how the cohomology of algebraic structures can be used to introduce the underlying geometry, which is not seen at the representation theoretic level, into the picture. The PI has been actively promoting the working knowledge of cohomological methods in representation theory and has recently organized several conferences in the area with an emphasis toward the development of junior mathematicians. The PI also co-directs a research group in algebra at the University of Georgia. This group provides practical training in representation theory for postdoctoral fellows and students through the use of computer algebra packages and the publication of research.

首席研究者（PI）将研究涉及与代数群，量子组和谎言级别代表理论的共同体学方法的问题。更具体地说，PI旨在扩大我们对代数组，量子组和谎言代数的重要共同学计算的了解。通过一系列涉及Lie代数和Frobenius kernel共同体学的步骤，研究国旗品种的线条捆绑捆绑会在这些计算中发挥着重要作用。 PI建议利用几何方法通过使用支持品种来研究表示理论。对于Lie Superalgebras，这种方法提供了对缺陷和非典型性良好组合概念的精美同源解释。代表理论大约在100年前出现了Frobenius和Schur的开创性工作，并且由于其与组合，代数几何形状，数字理论以及与物理学的应用有关，因此已成为数学的中心领域。拓扑家在整个20世纪开发了共同的理论，以构建代数不变性，以研究流形和拓扑空间。还针对诸如组之类的代数结构和代数来确定可以将其表示形式粘合在一起的方式定义了共同体。更引人注目的是，如何使用代数结构的共同体来引入潜在的几何形状，该几何形状在代表理论层面没有看到图片。 PI一直在积极促进代表理论中的共同方法的工作知识，并最近在该地区组织了几个会议，重点是发展初级数学家。 PI还与佐治亚大学代数的一个研究小组共同指导。该小组通过使用计算机代数软件包和研究发表的计算机代数软件包，为博士后研究员和学生提供代表理论的实用培训。