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计划在数学领域进行研究，该领域是在谎言代数和量子组，群集代数和非共同代数几何形状的表示理论的界面上进行研究。 谎言代数和量子群的表示理论是一个动态发展的现代数学领域。 它在其他数学领域以及物理学方面产生了很大的影响。 该提案的一个核心主题是理解和研究规范和晶体基础的特性，这些特性赋予了某些基本代数（例如量化的包围代数）具有额外的结构，这些结构已证明是为了理解这些代数及其作为矩阵的实现而有效的效率。 了解这些基础与紧密相关的结构之间的关系，例如完全积极的品种，几何晶体和群集代数，它们的量子和完全非共同的类似物是该提案的统一主题。在一个项目中，PI在一个项目中提出了一种基于Hecke代数的新方法，该方法基于Hecffef the HopfraS ane Hepbras：Alge al ealbras：ane hecfef al ealbras：代数并同时共同攻击它们。 这项研究产生的新信息允许构建新的解决方案，从而延符阳式方程。 该提案中的另一个项目是探索量化的包围代数的新基础，该代数具有显着的特性，例如编织组的行动以及与约瑟夫对量化包围代数的分解的兼容性。 预计这些新基础将解决分解代表的内态代数的问题，并有助于明确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通性群集结构，以及为这些代数的量子Gelfand-kirillov猜想的证明。