Quantum Affine Algebras: BGG reciprocity, Macdonald Polynomials, Schur postivity

该建议是关于仿射代数的表示理论之间的相互作用，其标准的最大抛物子代数，即当前代数，和量子仿射代数相关的一个简单的李代数。 它侧重于研究家庭的无限维和零级表示为每个这些代数和发展的连接与麦克唐纳多项式，Demazure字符和舒尔积极性。 其中一个目标是建立一个伯恩斯坦-Gelfand-Gelfand型对偶原则，这些类别和调查的组合和同调的后果，有这样一个对偶。 这个建议的另一个目标是发展量子仿射代数的无限维表示范畴的同调性质与这个范畴的张量结构之间的联系。 这种联系的存在是意想不到的，有点神秘，是由PI最近的一些工作提出的。 它只存在于量子水平，对这一点的深入理解应该对这一类别的研究产生重大影响。仿射李代数及其量子类似物的研究长期以来与许多不同的领域有着显着的联系，包括弦理论，共形场论，拓扑场论，无限维几何和数学物理。 该项目的许多主题都是由可解晶格模型中出现的问题所激发的。 仿射李代数和标准极大抛物子代数的表示将捕获重要的物理信息。 该项目还将提供一个表示理论框架，以了解各种组合问题。

The proposal is on the interplay between the representation theory of affine algebras, its standard maximal parabolic subalgebra, namely the current algebra, and the quantum affine algebra associated to a simple Lie algebra. It focuses on the study of families of infinite-dimensional and level zero representations for each of these algebras and develops connections with Macdonald polynomials, Demazure characters and Schur positivity. One of the goals is to establish a Bernstein-Gelfand-Gelfand type duality principle for these categories and to investigate the combinatorial and homological consequences of having such a duality. Another goal of this proposal is to develop a connection between the homological properties of the category of infinite-dimensional representations of the quantum affine algebra and the tensor structure of this category. The existence of such a connection is unexpected and somewhat mysterious and is suggested by some recent work of the PI. It exists only at the quantum level and a deeper understanding of this should have a substantial impact on the study of this category. It should also also yield connections with recent work of others on cluster algebras and categorification.The study of affine Lie algebras and their quantum analogs have long had remarkable connections to a number of different fields including string theory, conformal field theory, topological field theory, infinite dimensional geometry and mathematical physics. Many of the themes of the project are motivated by questions arising in solvable lattice models. Representations of affine Lie algebras and the standard maximal parabolic subalgebras will capture important physical information. The project will also provide a representation theoretic framework in which to understand various combinatorial problems.