Beyond Kirillov--Reshetikhin modules: character formulae and highest weight categories

该提案的主要主题是研究仿射李代数的表示理论与其量子类似物和其他代数结构之间的联系。仿射代数的有限维表示理论表现出许多与模表示理论相同的性质，并且非常复杂。PI的目的是证明人们可以将拟遗传代数与这些范畴联系起来，并探索证明这些表示的类似于著名的BGG对偶的可能性。这些想法中的许多也适用于更一般的扩展仿射李代数。该项目的另一个主题是了解量子仿射代数和簇代数表示之间的关系。其中一个重要的概念是素数表示和极小仿射的思想，PI计划对这些思想发展更深的理解。仿射李代数和它们的量子类似物与许多不同的领域有着显著的联系，包括弦论、保形场理论、拓扑场理论、无限维几何和数学物理。重要的物理现象可以通过建立在它们之上的数学理论得到更好的解释和预测。仿射李代数及其量子类比的表示将捕捉重要的物理信息，是建立上述与其他领域的联系的核心。这个项目的主要主题是研究仿射李代数及其量子模拟的表示理论。

The main theme of the proposal is to study connections between the representation theory of affine Lie algebras and their quantum analogs and other algebraic structures. The theory of finite dimensional representations of affine algebras shows many of the same properties as modular representation theory and is remarkably complex. The PI intends to show that one can associate quasi-hereditary algebras to such categories and to explore the possibility of proving an analog of the famous BGG duality for these representations. Many of these ideas will also work in the greater generality of extended affine Lie algebras. Another theme of the project is to understand the relationship between representations of quantum affine algebras and cluster algebras. An important concept in this is the idea of prime representations and minimal affinizations and the PI plans to develop a deeper understanding of these ideas. Affine Lie algebras and their quantum analogs have remarkable connections to a number of different fields including string theory, conformal field theory, topological field theory, infinite dimensional geometry and mathematical physics. Important physical phenomena can be better explained and predicted via mathematical theories built on them. Representations of affine Lie algebras and their quantum analogs will capture important physical information and are the core of the building the above connections to other fields. The main theme of this project is to study the representation theory of affine Lie algebras and their quantum analogs.