Crystals, level zero representations and the Littelmann path model

该项目解决了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推动的。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上的相应结果之间建立严格的联系。这涉及到研究仿射代数、密切相关的电流代数的有限维表示的范畴，以及这些代数对于量子参数的通用值的量子类比。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。这是研究Weyl模和本项目承担的不可约表示之间的扩张的一个重要动机。Chariand Pressley对Weyl模及其维度的猜想的研究，与Feigin和Loktevon的猜想有关。Feigin和Loktevon的猜想是有限维不可约李代数的融合积，是由他们对保形场论的研究而来的。期望关于Weyl模及其商Kirrillov/Reshetikhin模的结果将对Feigin和Loktev的猜想和构造提供进一步的见解和推广。一个重要的公开问题是为量子仿射代数的不可约有限维表示确定一个类似于WeylCharacter公式的特征标公式。第一步是研究Kirrillov/Reshetikhin模的这个问题，该项目通过观察Dorey在他对仿射Toda场论的研究中提出的一个猜想是否对这些模是正确的来进行这项工作。仿射李代数和量子代数的表示理论是一个非常活跃的研究领域。它对其他数学分支，如数论、纽结理论、组合学等都产生了重大影响。它与数学物理、仿射Toda场论以及统计力学中的不可解模型有着卓有成效的相互作用，在这些模型中，仿射代数表示的张量积猜想地描述了粒子的相互作用或融合。Pi的研究应该证实其中的一些理论，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也有应用。

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.