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

这一建议是关于仿射代数的表示理论，它的标准极大抛物子代数，即当前代数，和与单李代数相关的量子仿射代数之间的相互作用。重点研究了这些代数的无穷维零级表示族，并与Macdonald多项式、Demazure特征标和Schur正性建立了联系。目标之一是为这些范畴建立Bernstein-Gelfand-Gelfand型对偶原理，并研究具有这种对偶的组合和同调后果。这个建议的另一个目的是在量子仿射代数的无限维表示范畴的同调性质和这个范畴的张量结构之间建立联系。这种联系的存在是意想不到的，也有些神秘，PI最近的一些工作表明了这一点。它只存在于量子层面，对此的深入理解应该会对这一范畴的研究产生实质性的影响。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著的联系，包括弦论、保形场论、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驱动的。仿射李代数和标准极大抛物子代数的表示将捕捉到重要的物理信息。该项目还将提供一个理解各种组合问题的表征理论框架。

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.