Representation Theory of W-Algebras, Rational Cherednik Algebras, and Quantized Quiver Varieties

这个研究项目涉及表征理论，这是研究线性对称性的数学分支。对称通常形成代数结构，如结合式代数。本项目所研究的代数都具有几何性质，并且起源于量子物理学。目标是理解给定代数的表示，也就是说代数可以被表示为其他东西的线性对称。这里的基本构建块是不可约表示，这个项目的很大一部分涉及描述可能的不可约表示和计算重要的不变量，比如它们的维数。在这个项目中要解决的第二个重要问题是理解如何从不可约的表示中构建更复杂的表示。本课题主要研究表征理论w -代数、有理Cherednik代数和量子化颤振变数。对于w -代数，研究者计划对有限维不可约表示进行分类。对于有理Cherednik代数，研究者计划在给定支持下计算相关类别O中简单对象的数量。对于量子化的颤振变体，研究者计划计算有限维不可约表示的特征，研究相应类别O的结构，并为标准分层结构产生作为Ringel对偶的派生等价。这包括计算不可约模块的支撑尺寸。

This research project deals with representation theory, the branch of mathematics concerned with studying linear symmetries. Symmetries often form algebraic structures, such as associative algebras. The algebras studied in this project all have a geometric nature and originate from quantum physics. The goal is to understand representations of a given algebra, meaning ways the algebra can be represented as linear symmetries of something else. The basic building blocks here are irreducible representations, and a large part of this project deals with describing possible irreducible representations and computing important invariants, such as their dimensions. A second important question to be addressed in this project is understanding how more complicated representations may be built from irreducible representations. The project is devoted to the study of the representation theory W-algebras, rational Cherednik algebras, and quantized quiver varieties. For W-algebras, the investigator plans to classify the finite dimensional irreducible representations. For rational Cherednik algebras, the investigator plans to compute the number of simple objects in associated category O with given support. For quantized quiver varieties, the investigator plans to compute the characters of finite dimensional irreducible representations, study the structure of the corresponding category O, and produce derived equivalences as Ringel dualities for standardly stratified structures. This includes computing the dimensions of supports of irreducible modules.