DFG-RSF: Geometry and representation theory at the interface of Lie algebras and quivers

DFG-RSF：李代数和箭袋接口的几何和表示论

基本信息

批准号：
308831127
负责人：
Professor Dr. Markus Reineke
金额：
--
依托单位：
Russian Science Foundation
依托单位国家：
德国
项目类别：
Research Grants
财政年份：
2016
资助国家：
德国
起止时间：
2015-12-31 至 2018-12-31
项目状态：
已结题

来源：
https://gepris.dfg.de/gepris/projekt/308831127?language=en
关键词：
DFG RSF Geometry representation theory

项目摘要

The aim of the project is the study of various objects and structures endowed with large Lie groups of symmetries and linked to the geometric representation theory of quivers. The central innovative approach is to interleave the machinery of quiver representation theory and the methods and approaches of Lie theory in order to describe and study such objects, which are hard to deal with within one of the theories alone. The objects we are going to study include flag varieties and their degenerations, quiver Grassmannians, spherical and toric varieties, representations of finite and infinite-dimensional Lie algebras and their characters, and cyclic representations of abelian and contracted Lie algebras.The main objectives are:Description of toric degenerations of flag varieties and links with Newton-Okounkov theory, Study of the geometry of type A degenerate flag varieties and quiver Grassmannians, Description of PBW-type filtrations and associated graded spaces on highest weightrepresentations of simple and affine Kac-Moody Lie algebras, Development of a theory of quantum PBW filtrations, Description of algebro-geometric properties of finite and affine degenerate flag varieties, Study of the global geometry of the universal linear degeneration of type A flag varieties, Description of the structure of actions of Borel subalgebras and subgroups on naturalrepresentations and varieties, Description of graded characters of cyclic representations of current and affine algebras in terms of Macdonald polynomials.

该项目的目的是研究具有大型李对称群并与箭袋几何表示理论相关的各种物体和结构。核心创新方法是将颤抖表示理论的机制与李理论的方法和途径交织在一起，以描述和研究此类对象，而这些对象很难单独在一种理论中处理。我们要研究的对象包括旗形簇及其退化、颤动格拉斯曼函数、球面和环面簇、有限和无限维李代数的表示及其特征、阿贝尔和契约李代数的循环表示。主要目标是：旗形品种环面退化的描述及其与牛顿-奥孔科夫理论的联系，A型简并旗形品种和箭袋的几何研究Grassmannians，简单和仿射 Kac-Moody Lie 代数最高权重表示的 PBW 型过滤和相关分级空间的描述，量子 PBW 过滤理论的发展，有限和仿射简并旗簇的代数几何性质的描述，研究A 型标志变体的通用线性退化的全局几何，Borel 子代数和子群在自然表示上的作用结构的描述和变体，用麦克唐纳多项式描述当前代数和仿射代数的循环表示的分级特征。