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.

首席研究员（PI）将研究涉及代数对象的表示与其底层几何结构之间的联系的问题。PI计划研究的基本代数结构是李超代数、代数/有限群、量子群和Frobenius核。代数对象具有具体的（离散的）实现，并且通常在派生层次上产生潜在的丰富几何结构。上同调方法对揭示这个几何图形很有用。PI提出了使用涉及Balmer谱的新方法来描述李超代数的同调性质。他还计划计算代数和量子群的支持变体，作为连接几何对象和表征理论的一种方式。PI计划使用几何结构来理解有限群和代数群上同调的行为。众所周知，群、环、李代数和李超代数等代数结构在科学中自然地表现出来。对这些物体的基本认识在物理和化学中有许多不同的应用。这些结构通常很复杂。通常需要代数和几何方法来提取这些代数对象中的重要编码信息。就更广泛的影响而言，PI在全国范围内积极促进研究和教育的结合。他将继续指导美国国家科学基金会资助的佐治亚大学（UGA）研究与教育垂直整合项目。他也是弗吉尼亚大学维格尔代数小组的共同组织者，该小组为博士后和研究生提供当代数学方面的实践培训。PI将继续组织代数会议，重点关注初级数学家的发展，并将在美国和国外的研讨会、讲习班和暑期学校作为受邀演讲者，推广上同论和表示理论的工作知识。