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李群、量子群、交换和非交换簇代数以及非交换代数几何表示论的十字路口领域。 结合相关簇结构的研究，提出了一种研究各种量子代数中规范基的新方法。出现规范基的量子代数类包括还原代数群和双布鲁哈特晶胞的坐标环以及区间代数和汉克尔代数等新对象。 提出者最近发现，大多数上述簇和量子簇结构都允许完全不可交换的类似物，这一发现一方面导致了康采维奇簇猜想的证明，另一方面提供了从理性代数几何到其纯粹的非交换几何的新转变。研究的另一个途径，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情况也会带来新的幂零李代数和量子超代数是这些李代数的包络代数和对称代数的平面变形。这项研究的结果将应用于解决诸如计算还原群表示的对称幂的重数、计算相应旗簇和格拉斯曼量的上同调的舒伯特类乘积、构造新的全正簇、可积系统等问题。和赫克型代数，以及相关组合和几何结构的解释和阐述，包括晶体基底的“几何提升”作为理解当地朗兰兹的新工具 一致。 表示论是现代数学发展最活跃的领域之一。它对数学的其他领域和其他自然科学的众多应用都有很大的影响。匿名基和晶体基的概念对于表示论非常重要：仅仅建立这些基的存在性就有助于解决经典的枚举问题，例如计算不可约表示的重数或分解不可约表示的张量积。提议者发现的一类新的规范基有望解决分解表示的对称幂的老问题。因此，任何关于规范或晶体基础的信息都对表示论非常有益。提议者和其他研究人员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：完全正簇、几何晶体、交换簇、量子簇和完全非交换簇簇。理解规范基础背后的这些结构之间的关系是该提案的主要优先事项之一。这种关系已被证明是研究朗兰兹对应关系的有用工具——朗兰兹对应关系是 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关系。