Representation Theory, Cluster algebras, and Canonical Bases

表示论、簇代数和规范基

• 批准号: 1403527
• 负责人: Arkady Berenstein
• 金额: $ 16.83万
• 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: 2014
• 资助国家: 美国
• 起止时间: 2014-06-01 至 2017-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1403527&HistoricalAwards=false
• 关键词: Representation; Theory; Cluster; algebras; Canonical

The PI plans to conduct research in an area of mathematics at the interface of the representation theory of Lie algebras and quantum groups, cluster algebras, and noncommutative algebraic geometry. The representation theory of Lie algebras and quantum groups is a dynamically developing field of modern mathematics. It has had a large impact in other areas of mathematics, as well as in physics. A central theme of this proposal is to understand and investigate the properties of canonical and crystal bases, which endow certain fundamental algebras (such as quantized enveloping algebras) with an additional structure that has proven to be fruitful for understanding these algebras and their realizations as matrices. Understanding the relationship between these bases and closely related structures, such as totally positive varieties, geometric crystals, and cluster algebras, and their quantum and totally noncommutative analogues, is a unifying theme of this proposal.In one project the PI proposes a new approach to the study of Hecke algebras based on the discovery of some new Hopf algebras: the Hecke-Hopf algebras which contain Hecke algebras and co-act on them at the same time. The new information resulting from this study allows for the construction of new solutions to the quantum Yang-Baxter equation. Another project in the proposal is to explore new bases in quantized enveloping algebras which possess remarkable properties such as braid group action and the compatibility with Joseph's decomposition of the quantized enveloping algebras. These new bases are expected to settle the problem of decomposing endomorphism algebras of representations and to help explicitly compute the center of the ambient algebra. New information resulting from this study will be applied to constructing new quantum and noncommutative cluster structures and to the proof of the quantum Gelfand-Kirillov conjecture for those algebras.

PI 计划在李代数和量子群、簇代数和非交换代数几何表示论的接口的数学领域进行研究。 李代数和量子群的表示论是现代数学的一个动态发展的领域。 它对数学的其他领域以及物理学产生了巨大的影响。 该提案的中心主题是理解和研究规范基和晶体基的属性，这些属性赋予某些基本代数（例如量化包络代数）额外的结构，该结构已被证明对于理解这些代数及其作为矩阵的实现是富有成效的。 理解这些基和密切相关的结构（例如完全正簇、几何晶体和簇代数及其量子和完全非交换类似物）之间的关系是该提案的统一主题。在一个项目中，PI 提出了一种新方法基于一些新的 Hopf 代数的发现的 Hecke 代数的研究：包含 Hecke 代数并同时共同作用于它们的 Hecke-Hopf 代数 时间。 这项研究产生的新信息可以为量子杨-巴克斯特方程构建新的解。 该提案中的另一个项目是探索量化包络代数中的新基，这些基具有显着的性质，例如辫群作用以及与量化包络代数的 Joseph 分解的兼容性。 这些新基有望解决分解表示的自同态代数的问题，并帮助显式计算环境代数的中心。这项研究产生的新信息将应用于构建新的量子和非交换簇结构，并证明这些代数的量子格尔凡德-基里洛夫猜想。