Representation theory of W-algebras, quantum groups, symplectic reflection algebras and quantum Hamiltonian reductions

W-代数、量子群、辛反射代数和量子哈密顿量约简的表示论

• 批准号: 1161584
• 负责人: Ivan Loseu
• 金额: $ 13万
• 依托单位: Northeastern University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-09-01 至 2016-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1161584&HistoricalAwards=false
• 关键词: Representation; theory; algebras; quantum; groups

This project studies the representation theory of several different yet related associative algebras: finite W-algebras, symplectic reflection algebras, quantum groups and quantum Hamiltonian reductions associated to quivers. The investigator plans to classify finite dimensional irreducible modules over W-algebras and cyclotomic rational Cherednik algebras. He is going to relate various categories of representations of the universal enveloping algebras of semisimple Lie algebras and of W-algebras and use this relation to compute the dimensions of irreducible W-algebra modules. Next, the investigator will study a connection between W-algebras and quantum groups at a root of unity. Another related topic is the study of Harish-Chandra bimodules over symplectic reflection algebras and quantum groups at roots of unity. The investigator also plans to work on a conjecture of Rouquier describing the multiplicities in the categories O and a conjecture of Etingof on counting finite dimensional irreducible modules over symplectic reflection algebras. The latter will be approached in a more general context of quantum Hamiltonian reductions corresponding to Nakajima quiver varieties. The area of this project is Representation theory. Roughly speaking, Representation theory deals with symmetry, in particular, coming from Quantum Physics. Symmetries are thought as algebraic structures such as groups or algebras. The main problem is therefore is to understand how a given algebraic structure can be represented as a symmetry of some other objects, usually vector spaces. The algebraic structures studied in this project are certain associative algebras mostly arising in Quantum Mechanics: finite W-algebras, symplectic reflection algebras or quantum groups. Mostly, the project concentrates on a fundamental representation-theoretic problem - understanding basic, so called "irreducible"representations that serve as building blocks for more general ones with an emphasis on finite dimensional representations. Problems to be studied include computing the number of such representations, classifying them, computing their dimensions or finer invariants, called characters.

这个项目研究几个不同但相关的结合代数的表示理论：有限W-代数，辛反射代数，量子群和与箭图相关的量子哈密顿约化。研究者计划对W-代数和分圆有理Cherednik代数上的有限维不可约模进行分类。他将涉及各种类别的代表性的普遍包络代数的半单李代数和W-代数，并利用这种关系来计算尺寸的不可约W-代数模块。 接下来，研究者将研究W-代数和量子群在单位根上的联系。另一个相关的主题是辛反射代数和单位根量子群上的Harish-Chandra双模的研究。研究者还计划研究Rouquier的一个猜想，描述O类中的多重性，以及Etingof关于辛反射代数上有限维不可约模的计数的猜想。后者将在一个更一般的背景下，量子哈密顿减少相应的Nakajima的变种。该项目的领域是表征理论。粗略地说，表示论处理对称性，特别是来自量子物理学。对称性被认为是代数结构，如群或代数。因此，主要的问题是理解如何将给定的代数结构表示为其他对象（通常是向量空间）的对称性。在这个项目中研究的代数结构是一些主要出现在量子力学中的结合代数：有限W-代数，辛反射代数或量子群。大多数情况下，该项目集中在一个基本的表示理论问题-理解基本的，所谓的“不可约”表示，作为更一般的，重点是有限维表示的构建块。要研究的问题包括计算这些表示的数量，对它们进行分类，计算它们的维数或更精细的不变量，称为字符。