W-algebras and algebraic group actions

W-代数和代数群作用

• 批准号: 0900907
• 负责人: Pavel Etingof
• 金额: $ 13.78万
• 依托单位: Massachusetts Institute of Technology
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-06-01 至 2013-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0900907&HistoricalAwards=false
• 关键词: algebras; algebraic; group; actions

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).This project proposes research on two subjects: 1) Representation theory of W-algebras, 2) uniqueness properties for algebraic group actions. W-algebras (of finite type) are certain finitely generated associative algebras associated with nilpotent elements in semisimple Lie algebras. They originate from the work of B. Kostant of late 70's. In 90's they were studied by physicists. Starting from 2000 they attracted lot of attention of specialists in Representation Theory: Brundan, Ginzburg, Kleshchev, Premet, and others. In two recent years the investigator discovered a completely new approach to W-algebras based on Deformation quantization. This new approach allowed to him to prove many conjectures on W-algebras (mostly due to Premet) and, in particular, obtain the classification of their irreducible finite dimensional modules. The investigator plans to continue the study of representations of W-algebras and their q-deformations. In particular, he plans to prove a conjecture of Brundan-Goodwin- Kleshchev on the structure of the category O of W-algebras. Algebraic transformation group theory is a classical topic of algebraic geometry and group theory. One of major developments in algebraic transformation groups in recent 25 years is the theory of spherical varieties developed by Brion, Knop, Luna, Panyushev, Vinberg, Vust, and others. Spherical varieties are a particularly nice class of varieties equipped with a reductive group action. When the group is a torus, spherical is the same as toric. One of the nice features of spherical varieties is that their classification may be obtained in entirely combinatorial terms. In the recent few years the investigator obtained certain uniqueness properties of spherical varieties in terms of their combinatorial invariants proving conjectures due to Brion, Knop and Luna. The investigator plans to generalize these results to arbitrary varieties equipped with an action of a reductive group. In particular, he plans to prove that a smooth affine G-variety is uniquely determined by its algebra of U-invariants.This research projects deals with different kinds of symmetries arising both in pure mathematics and in physics. For instance, W-algebras are certain algebraic structures appeared in pure algebraic studies of Kostant in late 70's. Since then they found a number of applications in representation theory. On the other hand they are a manifestation of the notion of W-symmetry from Conformal field theory extensively studied by physicists. So the investigator's research project will contribute to pure mathematics and may have some applications to physics. The second part of this research project deals with a more classical notion of symmetries coming from geometry.

该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。该项目提出了两个主题的研究：1）W-代数的表示理论，2）代数群作用的唯一性。W-代数（有限型）是半单李代数中与幂零元相关联的一类非生成结合代数。它们源于B的工作。70年代末的科斯坦特。90年代，物理学家开始研究它们。从2000年开始，他们吸引了很多专家的注意力在表示理论：布伦丹，金兹伯格，Kleshchev，普雷梅特，和其他人。在最近的两年里，研究者发现了一种基于形变量子化的W-代数的全新方法。这种新的方法使他能够证明许多progratures对W-代数（主要是由于普雷梅），特别是，获得分类的不可约有限维模块。研究人员计划继续研究W-代数的表示及其q-变形。特别是，他计划证明一个猜想的布伦丹-古德温- Kleshchev的结构的范畴O的W-代数。代数变换群论是代数几何和群论的经典课题。球面簇理论是近25年来代数变换群理论的一个重要发展，它是由Brion，Knop，Luna，Panyushev，Vinberg，Vust等人提出的。球形变种是一类特别好的变种，具有还原性基团作用。当群是环面时，球面与环面相同。球形变种的一个很好的特点是它们的分类可以完全用组合术语来获得。近年来，研究者利用球簇的组合不变量得到了球簇的某些唯一性性质，证明了Brion，Knop和Luna等人的定理。研究人员计划将这些结果推广到任意品种配备了一个行动的还原组。特别是，他计划证明一个光滑仿射G-簇是唯一确定的代数的U-不变量。这个研究项目涉及不同种类的对称性都出现在纯数学和物理学。例如，W-代数是70年代后期Kostant在纯代数研究中出现的一类代数结构。从那时起，他们发现了一些应用在代表性理论。另一方面，它们是物理学家广泛研究的共形场论中W对称性概念的体现。因此，研究者的研究项目将有助于纯数学，并可能在物理学中有一些应用。这个研究项目的第二部分涉及来自几何的对称性的一个更经典的概念。