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

这项研究项目涉及表示论，这是数学的一个分支，致力于研究线性对称。对称通常形成代数结构，例如结合代数。这个项目中研究的代数都具有几何性质，都起源于量子物理。目标是理解给定代数的表示，这意味着该代数可以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，本项目的很大一部分内容涉及描述可能的不可约表示和计算重要的不变量，例如它们的维度。这个项目要解决的第二个重要问题是理解如何从不可约表示建立更复杂的表示。该项目致力于研究表示论W-代数、有理Cherednik代数和量子化箭图簇。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，研究者计划计算具有给定支撑度的关联范畴O中的简单对象的数目。对于量化箭图簇，研究者计划计算有限维不可约表示的特征，研究相应范畴O的结构，并产生与标准分层结构的Ringel对偶相同的导出等价。这包括计算不可约模的支承的维度。

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.