Crystals, level zero representations and the Littelmann path model

晶体、零级表示和 Littelmann 路径模型

• 批准号: 0500751
• 负责人: Vyjayanthi Chari
• 金额: $ 12.5万
• 依托单位: University of California-Riverside
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-08-01 至 2009-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0500751&HistoricalAwards=false
• 关键词: Crystals; level; zero; representations; Littelmann

The project addresses questions in the representation theoryof affine Lie algebras and their quantum analogs. Theproblems to be studied are motivated by applications tomathematical physics and combinatorics. One of the goals ofthe project is to make a rigorous connection betweencombinatorial results on crystals and the correspondingresults on representations of the affine Lie algebra. Thisinvolves studying the category of finite dimensionalrepresentations of the affine algebra, the closely relatedcurrent algebra and the quantum analogs of these algebras forgeneric values of the quantum parameter. This category ofrepresentations is not semisimple and exhibits featuressimilar to modular representation theory. This is animportant motivation for the study of Weyl modules and ofextensions between irreducible representations of thesealgebras undertaken by the project. The study of Weylmodules and the conjecture on their dimension made by Chariand Pressley is related to the conjectures of Feigin and Loktevon the fusion product of finite dimensional irreduciblerepresentations of a simple Lie algebra coming from theirstudy on conformal field theory . It is expected that theresults on the Weyl modules and their quotients theKirrillov/Reshetikhin modules will provide further insightinto and also lead to generalizations of the conjectures andconstructions of Feigin and Loktev. An important open problem is to determine a character formula for the irreduciblefinite dimensional representations of the quantum affine algebras analogous to the Weylcharacter formula. A first step is to study this problem forthe Kirrillov/Reshetikhin modules and the project pursues this byseeing if a conjecture of Dorey made in his study of affineToda field theories is correct for these modules.The representation theory of affine Lie algebras and thequantum algebras is an area where there is intense researchactivity. It has had significant impact on other branches ofmathematics such as number theory, knot theory, combinatoricsto name a few. It has had fruitful interaction withmathematical physics, in affine Toda field theories, and insolvable models in statistical mechanics where the tensorproduct of representations of affine algebras conjecturallydescribes the interactions or fusing of particles. The PI'sstudy should confirm some of these theories and also haveapplications in the representation theory of affine Liealgebras of positive level and vertex algebra constructions ofsuch representations.

该项目解决了仿射李代数及其量子类似物的表示论问题。 要研究的问题是由数学物理和组合数学的应用激发的。 该项目的目标之一是在晶体的组合结果与仿射李代数表示的相应结果之间建立严格的联系。 这涉及研究仿射代数的有限维表示的范畴、密切相关的电流代数以及这些代数对于量子参数的通用值的量子模拟。这类表示不是半简单的，并且表现出类似于模表示理论的特征。 这是该项目研究 Weyl 模以及这些代数的不可约表示之间的扩展的重要动机。 Chariand Pressley对Weyl模的研究及其维数猜想与Fegin和Loktevon在共形场论研究中提出的简单李代数的有限维不可约表示的融合积的猜想有关。预计 Weyl 模块及其商、Kirrillov/Reshetikhin 模块的结果将提供对 Feigin 和 Loktev 的猜想和构造的进一步洞察，并导致其概括。一个重要的开放问题是确定类似于 Weylcharacter 公式的量子仿射代数的不可约有限维表示的特征公式。第一步是研究 Kirrillov/Reshetikhin 模块的这个问题，该项目通过查看 Dorey 在仿射 Toda 场论研究中所做的猜想对于这些模块是否正确来实现这一目标。仿射李代数和量子代数的表示论是一个研究活跃的领域。它对数学的其他分支产生了重大影响，例如数论、结论、组合学等。它与数学物理学、仿射户田场论以及统计力学中的不可解模型进行了富有成效的相互作用，其中仿射代数表示的张量积推测地描述了粒子的相互作用或融合。 PI的研究应该证实其中一些理论，并且在正级仿射李代数表示论和此类表示的顶点代数构造中也有应用。