Cohomology, Geometry and Representation Theory: Algebraic Groups, Quantum Groups and Lie Superalgebras

上同调、几何和表示论：代数群、量子群和李超代数

• 批准号: 1002135
• 负责人: Daniel Nakano
• 金额: $ 17万
• 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: 2010
• 资助国家: 美国
• 起止时间: 2010-08-15 至 2014-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1002135&HistoricalAwards=false
• 关键词: Cohomology; Geometry; Representation; Theory; Algebraic

The Principal Investigator (PI) will investigate problems involving the connections between representations of algebraic objects and their underlying geometric structures. The basic algebraic structures that the PI proposes to study are Lie superalgebras, algebraic/finite groups, quantum groups, and Frobenius kernels. The algebraic objects have concrete (discrete) realizations, and often times the underlying rich geometric structures arise at the derived level. Cohomological methods are useful for unveiling this geometry. The PI proposes to use new methods involving the Balmer spectrum to describe homological properties of Lie superalgebras. He also plans to make calculations of support varieties for algebraic and quantum groups as a way to connect geometric objects and representation theory. The PI plans to use geometric structures to understand the behavior of the cohomology of finite and algebraic groups. It is well known that algebraic structures such as groups, rings, Lie algebras, and Lie superalgebras manifest themselves naturally in science. The basic understanding of these objects have been used in many different applications in physics and chemistry. These structures are often complicated.Both algebraic and geometric methods are often necessary to extract the important encoded information within these algebraic objects. In terms of broader impacts, the PI has been active nationally in the promotion of integrating research and education. He will continue to direct the NSF funded VIGRE (Vertical Integration of Research and Education) Program at the University of Georgia (UGA). He is also a co-organizer of the VIGRE Algebra Group at UGA which provides practical training in contemporary mathematics to postdoctoral fellows and graduate students. The PI will continue to organize conferences in algebra with an emphasis toward the development of junior mathematicians, and will promote the working knowledge of cohomology and representation theory as an invited speaker at seminars, workshops, and summer schools in the U.S. and abroad.

主要研究者（PI）将调查涉及代数对象表示及其潜在几何结构之间联系的问题。 PI提议研究的基本代数结构是超级甲壳虫，代数/有限基团，量子基团和Frobenius内核。代数对象具有具体（离散的）实现，并且通常在派生级别出现了基本的丰富几何结构。协同学方法可用于揭示此几何形状。 PI建议使用涉及Balmer频谱的新方法来描述Like Superalgebras的同源特性。他还计划计算代数和量子组的支持品种，以此来连接几何对象和表示理论。 PI计划使用几何结构来了解有限和代数群体的共同体的行为。众所周知，诸如群体，环，代数和谎言的代数结构在科学中自然而然地表现出来。这些对象的基本理解已用于物理和化学的许多不同应用中。这些结构通常是复杂的。代数和几何方法通常需要在这些代数对象中提取重要的编码信息。就更广泛的影响而言，PI在全国范围内一直活跃于促进研究和教育的促进。他将继续指导佐治亚大学（UGA）的NSF资助的Vigre（研究与教育计划的垂直整合）。他还是UGA的Vigre代数集团的共同组织者，该组织为博士后研究员和研究生提供了当代数学的实用培训。 PI将继续组织代数的会议，重点是为初级数学家的发展发展，并将作为在美国和国外的研讨会，研讨会和暑期学校的邀请发言人来促进同居和代表理论的工作知识。