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

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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.