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.

该项目解决仿射李代数及其量子类似物的表示理论中的问题。 所要研究的问题是由数学物理和组合学的应用所激发的。 该项目的目标之一是在晶体上的组合结果和仿射李代数表示的相应结果之间建立严格的联系。 这涉及到研究范畴的有限dimensionalrepresentations的仿射代数，密切相关的电流代数和量子类似物，这些代数的通用值的量子参数。这类表示不是半简单的，表现出类似于模表示理论的特征。 这是研究Weyl模和这些代数的不可约表示之间的扩张的一个重要动机 该项目承担。 Weyl模的研究及其维数猜想与单李代数的有限维不可约表示的融合积Feigin和Loktevon的构造有关 来自于他们对共形场论的研究。我们期望Weyl模及其等价物Kirrillov/Reshetikhin模的结果将进一步深化和推广Feigin和Loktev的结构和构造。一个重要的公开问题是确定量子仿射代数的不可约有限维表示的一个类似于Weyl特征标公式的特征标公式。第一步是研究这个问题的Kirrillov/Reshetikhin模块和该项目追求这一点，看看如果一个猜想的Dorey在他的研究仿射Toda场理论是正确的这些modules.The表示理论的仿射李代数和量子代数是一个领域，有激烈的研究活动。它对其他数学分支产生了重大影响，如数论、纽结理论、组合数学等。它与数学物理、仿射户田场论和统计力学中的不可解模型有着富有成效的相互作用，在统计力学中，仿射代数表示的张量积直观地描述了粒子的相互作用或融合。PI的研究应该证实了其中的一些理论，并在正水平仿射李代数的表示理论和这种表示的顶点代数构造中有应用。