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

本课题研究了有限w代数、辛反射代数、量子群和与颤振相关的量子哈密顿约化等几种不同但又相关的关联代数的表示理论。研究者计划在w -代数和切环有理Cherednik代数上对有限维不可约模进行分类。他将把半单李代数和w -代数的泛包络代数的各种表示联系起来，并利用这种联系来计算不可约w -代数模块的维数。接下来，研究者将研究w -代数和量子群之间的联系。另一个相关的课题是研究辛反射代数上的Harish-Chandra双模和单位根上的量子群。研究人员还计划研究Rouquier关于描述O类多重性的猜想和关于在辛反射代数上计数有限维不可约模的Etingof猜想。后者将在与中岛颤振变体相对应的量子哈密顿约简的更一般的背景下进行讨论。这个项目的领域是表征理论。粗略地说，表征理论处理的是对称性，尤其是来自量子物理学的对称性。对称被认为是代数结构，如群或代数。因此，主要的问题是如何理解一个给定的代数结构可以被表示为一些其他对象的对称，通常是向量空间。本课题研究的代数结构主要是量子力学中出现的一类关联代数：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，作为更一般的构建块，强调有限维表示。要研究的问题包括计算这种表示的数量，对它们进行分类，计算它们的维度或更精细的不变量，称为字符。

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.