Representation Theory, Quantum Groups, and Birational Algebraic Geometry

表示论、量子群和双有理代数几何

• 批准号: 0501103
• 负责人: Arkady Berenstein
• 金额: --
• 依托单位: University of Oregon Eugene
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-06-01 至 2008-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0501103&HistoricalAwards=false
• 关键词: Representation; Theory; Quantum; Groups; Birational

This project is devoted to investigation of the area lying at the crossroads of the representation theory of Lie groups, quantum groups, birational algebraic geometry, and piecewise-linear combinatorics. A new approach to the study of Lusztig's canonical bases and Kashiwara'scrystal bases is proposed, based on quantum cluster algebras and geometric crystals. New information resulting from this study will be applied to computing the multiplicities for the representations of reductive groups and for constructing new totally positive varieties. The results of this study will also be used for solving problems emerging in the representations of discrete subgroups of reductive algebraic groups as well as for explication and elaboration of related combinatorial and geometric structures. Representation theory of Lie algebras and quantum groups is one of the most dynamically developing fields of modern Mathematics. This theory has a large impact on other fields of Mathematics and generates numerous applications in other NaturalSciences. In their turn, the concepts of canonical and crystal bases are of great importance for the representation theory: a mere establishing of their existence has helped in solving classical enumeration problems (e.g., the problem of computing multiplicities of irreducible representations or decomposing tensor products of irreducible representations). Therefore, any information on canonical or crystal bases would be very beneficial for the representation theory. Understanding the relationship between the discrete (i.e., combinatorial) and continuous (i.e., geometric) structures of the canonical bases is one of the main priorities of this project. This relationship has proved to be a useful tool in the study of a famous Langlands correspondence -- the most mysterious and inspiring correspondence between Algebra and Geometry of the 20th century Mathematics.

该项目致力于调查位于谎言组，量子群，遗嘱代数几何形状和分段线性组合学的代表理论的区域。基于量子群集代数和几何晶体，提出了一种研究Lusztig规范底座和Kashiwara'scrystal碱基的新方法。这项研究产生的新信息将应用于计算还原群体表示的多重性，并构建新的完全积极的品种。 这项研究的结果还将用于解决还原代数基团的离散亚组以及相关组合和几何结构的解释和阐述中出现的问题。谎言代数和量子群的表示理论是现代数学最动态发展的领域之一。该理论对其他数学领域具有很大的影响，并在其他自然学院中产生了许多应用。反过来，对于表示理论而言，规范和晶体基础的概念非常重要：仅建立它们的存在就有助于解决经典的枚举问题（例如，计算不可减少表示形式的多重性或不可减免表示的张力张力产品的分解张量的问题）。因此，有关规范或晶体基础的任何信息对代表理论都非常有益。 了解该项目的主要优先事项之一是理解离散（即组合）与连续（即几何）结构之间的关系之一。事实证明，这种关系是研究著名的兰兰兹信件的有用工具，这是20世纪数学的代数和几何形状之间最神秘，最具启发性的对应关系。