Representation Theory, Cluster Algebras, and Canonical Bases

表示论、簇代数和规范基

• 批准号: 1101507
• 负责人: Arkady Berenstein
• 金额: $ 16.38万
• 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: 2011
• 资助国家: 美国
• 起止时间: 2011-09-01 至 2014-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1101507&HistoricalAwards=false
• 关键词: Representation; Theory; Cluster; Algebras; Canonical

The main theme of this proposal is to investigate the area lying at the crossroads of the representation theory of Lie groups, quantum groups, commutative and non-commutative cluster algebras, and non-commutative algebraic geometry. A new approach to the study of canonical bases in various quantum algebras is proposed along with the study of the related cluster structures. The class of quantum algebras in which the canonical bases arise include coordinate rings of reductive algebraic groups and double Bruhat cell as well as new objects such as interval and Hankel algebras. It has been recently discovered by the proposer that most of the above mentioned cluster and quantum cluster structures admit totally non-commutative analogues, which discovery, on the one hand, resulted in the proof of Kontsevich Cluster Conjecture and, on the other hand, provides a new transition from the rational Algebraic Geometry to its purely non-commutative counterpart. Another avenue of the research, quantum folding is a new approach to looking at the Dynkin symmetries of quantum groups that, quite surprisingly, produces the Langlands duals of the classical fixed points groups Even the simplest cases of quantum folding bring about new nilpotent Lie algebras and quantum uberalgebras that are flat deformations of both enveloping and symmetric algebras of those Lie algebras. The results of this study will be applied for solving problems such as computing the multiplicities for the symmetric powers of representations of reductive groups, computing products of Schubert classes in cohomology of the corresponding flag varieties and Grassmannians, constructing new totally positive varieties, integrable systems, and Hecke type algebras as well as for explication and elaboration of related combinatorial and geometric structures including the ``geometric lifting'' of crystal bases as a new tool in understanding the local Langlands correspondence. Representation theory is one of the most dynamically developing fields of modern Mathematics. It has a large impact in other fields of Mathematics and numerous applications in other Natural Sciences. The concepts of anonical and crystal bases are of great importance for the representation theory: a mere establishing of existence of such bases has helped in solving classical enumeration problems like computing multiplicities of irreducible representations or decomposing tensor products of irreducible representations. A new class of canonical bases discovered by the proposer is expected to settle an old problem of decomposing symmetric powers of representations. Therefore, any information on canonical or crystal bases would be very beneficial for the representation theory. The results of the proposer and other researchers suggest natural algebro-geometric counterparts for the purely discrete canonical and crystal bases: totally positive varieties, geometric crystals, commutative, quantum, and totally noncommutative cluster varieties. Understanding the relationship between these structures underlying the canonical bases is one of main priorities of this proposal. This relationship has proved to be a useful tool in the study of Langlands correspondence --the most mysterious and inspiring correspondence between Algebra and Geometry of the 20th and 21st century Mathematics.

该提案的主题是研究位于谎言群体，量子群，交换性和非交通性群集代数以及非共同代数几何形状的代表理论十字路口的区域。 提出了一种研究各种量子代数中规范碱基的新方法，并研究了相关群集结构的研究。出现规范碱基的量子代数类别包括还原代数基和双bruhat细胞的坐标环，以及新物体，例如Interval和Hankel代数。 提议者最近发现，上述大多数提到的群集和量子集群结构都承认完全非共同的类似物，一方面发现了肯特维奇集群猜想的证明，另一方面，它提供了从合理的代数质量到其纯粹的非commutical uncomporticational uncomporticational commuticational commuticationally commuticational commuticationally commuticationally commuticational uncompart。这项研究的另一个途径，量子折叠是一种新的方法来查看量子组的dynkin对称性，令人惊讶的是，这些量子群会产生经典固定点组的兰兰兹双重偶数，即使是最简单的量子折叠案例，也带来了新的nilpotent lie代数和量子Uberalgebras，这些lie uberalgebras既是这些易于体现和对称的e anderge and symemsmets ealbrass and symermetmetsmetmetsmetsmetsmetsmetsmetsmetsmetsmetmets themermets的折叠式折叠。这项研究的结果将应用于解决问题，例如计算还原群体表示的对称能力的多样性，计算相应国旗品种和硕士学位中舒伯特类别的产品的计算产品，构建了新的完全积极的品种，综合系统，综合系统，以及构建了相关的构造和熟悉的组合，并构建了相关的组合级别的构建。将晶体底座的提升为理解当地兰兰对应的新工具。 表示理论是现代数学最动态发展的领域之一。它在其他数学领域和其他自然科学领域的应用中具有很大的影响。对于表示理论而言，肛门和晶体基础的概念非常重要：仅建立这种基础的存在有助于解决经典的枚举问题，例如计算不可减少表示的多重性或分解不可减至表示的张力产品。预计提议者发现的一类新的规范基础将解决分解对称代表权的旧问题。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。提议者和其他研究人员的结果提出了纯粹的离散规范和晶体碱基的天然代数几何对应物：完全阳性的品种，几何晶体，交换性，量子，量子和完全非交通性簇品种。了解规范基础基础的这些结构之间的关系是该提议的主要优先事项之一。事实证明，这种关系是研究兰兰兹信函的有用工具，这是20世纪和21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。