The Quantum McKay Correspondence: Classifying "Finite Subgroups" of a Quantum Group with Graphs Open Access

Abstract

The McKay Correspondence classifies finite subgroups of the rotation group of 3-space via graphs. In this paper we explore a quantum version of this correspondence. Specifically, we will cover the needed background on category theory, vertex operator algebras, and quantum groups to explain a powerful technique used by Kirillov and Ostrik to develop a quantum analog to the McKay correspondence.

Table of Contents

1 Background 1

1.1  LiegroupsandLieAlgebras ........................... 1

1.2  RepresentationTheory .............................. 4

1.3  TheMcKayCorrespondence........................... 5

1.4  CategoryTheory ................................. 6

1.5  CategoricalGraphicalCalculus ......................... 10

1.6  QuantumGroups................................. 13

1.7  VertexOperatorAlgebras ............................ 15

2 Kirillov and Ostrick’s q-Analogue 16

2.1  Preliminaryresults ................................ 16

2.2  SanityCheck: RepresentationsofFiniteGroups . . . . . . . . . . . . . . . . 20

2.3  ResultsonVertexOperatorAlgebras ...................... 21

2.4  TheMainProof.................................. 23

2.4.1 Case:A .................................. 24 2.4.2 Case:D .................................. 24 2.4.3 Case:T .................................. 24 2.4.4 Case:E6.................................. 25 2.4.5 Case:E7.................................. 25 2.4.6 Case:E8.................................. 25 2.4.7 DiagramRepresentationComposition.................. 26

3 Conclusion 27 List of Figures

1  TheClassificationofSemisimpleLieAlgebras . . . . . . . . . . . . . . . . . 3

2  TheClassificationofAffineLieAlgebras .................... 5

3  TheExplicitformoftheMcKayCorrespondence . . . . . . . . . . . . . . . 6

4  Acommutingdigraminacategorywithzeromorphisms . . . . . . . . . . . 7

5  Dnwithneven.................................. 26

6  E8 ......................................... 26

7  E6 ......................................... 26

About this Master's Thesis

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School	Laney Graduate School
Department	Mathematics
Degree	M.S.
Submission	Master's Thesis
Language	English
Research Field	Mathematics
Keyword	McKay Correspondence Quantum Groups
Committee Chair / Thesis Advisor	Duncan, John, Emory University
Committee Members	Zureick-Brown, David, Emory University Weiner, Sharron, Emory University

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