CAREER: Geometric category O and symplectic duality

职业：几何范畴 O 和辛对偶性

• 批准号: 0950383
• 负责人: Nicholas Proudfoot
• 金额: $ 40万
• 依托单位: University of Oregon Eugene
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-06-15 至 2017-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0950383&HistoricalAwards=false
• 关键词: CAREER; Geometric; category; symplectic; duality

The purpose of this grant is to study the structure of certain categories of sheaves on symplectic algebraic varieties. When the variety in question is the cotangent bundle to a flag variety, then the category that one obtains is equivalent to a block of the "Category O" of representations of a Lie algebra, originally introduced by Bernstein, Gelfand, and Gelfand. For this reason, this category is referred to as "Geometric Category O". This category is conjectured to have many beautiful properties, which have been verified in certain special cases that arise naturally from representation theory and polyhedral geometry. Chief among these conjectures is that the categories are Koszul, and that each symplectic variety has a "symplectic dual" whose associated category is Koszul dual to that of the original variety. This phenomenon is expected to form a bridge between various approaches to categorification of Lie algebra representations and link invariants that were previously thought to be unrelated.Symplectic algebraic varieties arise naturally from many different areas of mathematics. Geometric and topological properties of hypertoric varieties have shed new light on the topology of hyperplane arrangements and the combinatorics of matroids. Quiver varieties provide geometric realizations of actions of infinite dimensional Lie algebras, leading to canonical bases and to actions on categories of sheaves. This grant plays a key role in this picture, providing new geometric insight to such phenomena as Gale duality in combinatorics and level-rank duality in representation theory. This project will contribute to each of these endeavors independently, and will also advance a common treatment that unifies our understanding of the various individual phenomena. In addition to the research component, the grant includes an annual workshop for graduate students and postdocs that bridges these various fields.

这个补助金的目的是研究辛代数簇上某些类别的层的结构。 当所讨论的簇是一个旗簇的余切丛时，则得到的范畴等价于李代数表示的“范畴O”的一个块，最初是由伯恩斯坦、Gelfand和Gelfand引入的。 因此，这个类别被称为“几何类别O”。 这个范畴被证明有许多美丽的性质，这些性质在某些特殊的情况下得到了验证，这些特殊的情况是由表示论和多面体几何自然产生的。 主要的这些假设是范畴是Koszul，并且每个辛簇都有一个“辛对偶”，其相关范畴是原始簇的Koszul对偶。 这一现象有望在李代数表示和链接不变量的各种分类方法之间形成一座桥梁，而这些方法以前被认为是不相关的。辛代数簇自然地出现在许多不同的数学领域。超环面簇的几何和拓扑性质为超平面排列的拓扑和拟阵的组合学提供了新的线索。箭图簇提供了无限维李代数作用的几何实现，导致了规范基和层范畴上的作用。这笔赠款在这幅图中起着关键作用，提供了新的几何洞察力，如盖尔对偶组合和等级对偶表示论等现象。这个项目将有助于每一个独立的这些努力，也将推进一个共同的治疗，统一我们对各种个别现象的理解。 除了研究部分，赠款还包括为研究生和博士后举办的年度研讨会，以弥合这些不同领域。