W-algebras and algebraic group actions

该奖项由《2009年美国复苏和再投资法案》(公法111-5)资助。本项目建议研究两个主题：1)W-代数的表示理论，2)代数群作用的唯一性性质。(有限类型的)W-代数是与半单李代数中的幂零元相联系的有限生成结合代数。它们起源于七十年代末S的工作，九十年代的S被物理学家研究。从2000年开始，它们吸引了许多表征理论专家的注意：布伦丹、金兹堡、克莱舍夫、普雷米特等。最近两年，研究者发现了一种基于形变量子化的全新的W-代数方法。这一新的方法使他能够证明许多关于W-代数的猜想(主要是由于Premet)，特别是得到了它们的不可约有限维模的分类。这位研究者计划继续研究W-代数的表示及其Q-变形。特别地，他计划证明Brundan-Goodwin-Kleshchev关于W-代数范畴O的结构的一个猜想。代数变换群论是代数几何和群论的经典课题。近25年来代数变换群的主要发展之一是由Brion，Knop，露娜，Panyushev，Vinberg，Vust等人发展起来的球体理论。球形变种是一类特别好的变种，具有还原基团作用。当群是环面时，球面与环面相同。球形变种的一个很好的特征是它们的分类可以完全用组合项来获得。最近几年，研究者利用球簇的组合不变量证明了Brion，Knop和露娜的猜想，得到了球簇的某些唯一性性质。研究人员计划将这些结果推广到带有还原群作用的任意变种。特别是，他计划证明光滑仿射G-簇是由它的U-不变量代数唯一确定的。这项研究项目涉及纯数学和物理学中出现的不同类型的对称性。例如，W-代数是70年代末S在Kostant的纯代数研究中出现的某种代数结构，从那时起，它们在表示论中得到了许多应用。另一方面，它们是物理学家广泛研究的共形场理论中W对称性概念的一种表现。因此，研究人员的研究项目将对纯数学做出贡献，并可能在物理学中有一些应用。这项研究项目的第二部分涉及一个更经典的对称性概念，它来自几何。

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.