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.

本提案的主要主题是研究位于李群，量子群，交换和非交换簇代数以及非交换代数几何的表示理论十字路口的区域。提出了一种研究各种量子代数正则基的新方法，并对相关的簇结构进行了研究。产生正则基的量子代数包括约化代数群的坐标环和双Bruhat单元，以及区间代数和Hankel代数等新对象。最近，提出者发现，上述的大多数簇和量子簇结构都承认完全非交换的类似物，这一发现一方面证明了Kontsevich簇猜想，另一方面提供了从有理代数几何到其纯粹非交换的对应物的一个新的过渡。另一种研究途径，量子折叠是一种观察量子群的Dynkin对称性的新方法，令人惊讶的是，它产生了经典不动点群的朗兰兹对偶，即使是最简单的量子折叠也会产生新的幂零李代数和量子超代数，它们是这些李代数的包络代数和对称代数的平面变形。本文的研究结果将应用于计算约化群的对称幂表示的多重性、计算相应的标志簇和Grassmannians的上同调中Schubert类的积、构造新的全正簇、可积系统、以及Hecke型代数以及相关组合和几何结构的解释和阐述，包括晶体碱的“几何提升”，作为理解局部朗兰兹对应的新工具。表征理论是现代数学中发展最活跃的领域之一。它对数学的其他领域产生了巨大的影响，并在其他自然科学中有许多应用。不规则基和晶体基的概念对表示理论具有重要意义：仅仅建立这些基的存在性就有助于解决经典的枚举问题，如计算不可约表示的复数或分解不可约表示的张量积。作者发现的一类新的正则基有望解决对称幂表示分解的老问题。因此，任何关于规范基或晶体基的信息都将对表征理论非常有益。该提议者和其他研究者的结果提出了纯离散正则基和晶体基的自然代数-几何对应物：完全正的变体、几何晶体、交换的、量子的和完全非交换的簇变体。理解这些结构之间的关系是规范基础的主要优先事项之一。这种关系被证明是研究朗兰兹对应的一个有用的工具——朗兰兹对应是20世纪和21世纪数学中代数和几何之间最神秘和最鼓舞人心的对应。