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.

该项目研究了几个不同但相关的联想代数的表示理论：有限的W-代数，符号反射代数，代数，量子群和与颤动相关的量子哈密顿减少。研究人员计划将有限的尺寸不可还原模块分类，并在W-代数和环状有理Cherednik代数上进行分类。他将与半膜的通用代数和w-Algebras的通用代数的各种表示形式联系起来，并使用此关系来计算不可约W-Algebra模块的尺寸。 接下来，研究人员将研究统一根部的W- algebras与量子组之间的联系。另一个相关的主题是对统一根部的象征反射代数和量子群的Harish-Chandra双模型的研究。研究者还计划对鲁奎尔的猜想进行构想，以描述o类别中的多重性，并猜想Etingof在计数有限的尺寸不可减至的模块上，而不是符号反射代数。后者将在与Nakajima Quiver品种相对应的量子哈密顿量减少的更一般环境中接近。该项目的领域是代表理论。粗略地说，表示理论涉及对称性，特别是来自量子物理学。对称性被认为是代数结构，例如组或代数。因此，主要问题是了解如何将给定代数结构表示为其他某些对象（通常是向量空间）的对称性。该项目中研究的代数结构是某些相关代数，主要是在量子力学中产生的：有限的W-代数，符号反射反射代数或量子组。大多数情况下，该项目集中于一个基本表示问题 - 理解基本的表达，所谓的“不可约”表示，这些表示是更笼统的构建模块，重点是有限的维度表示。要研究的问题包括计算此类表示形式的数量，对它们进行分类，计算其尺寸或更细的不变式，称为字符。