Endotrivial modules over preprojective algebras

Inhaltsverzeichnis (Table of Contents)

• Introduction
• Modules over preprojective algebras
  • Prerequisites and basic definitions.
  • Reflection functors.
  • Endotrivial modules over preprojective algebras of Dynkin type ADE
  • Classification of endotrivial modules over the preprojective algebra of type Dn
• Quivers with relations for symmetrizable Cartan matrices
  • Symmetrizable Cartan matrices and associated algebras
  • The Weyl group
  • Locally free H-modules
  • An analogy to the representation theory of modulated graphs
  • Representation theory of modulated graphs.
  • Reflection functors
• References
• Acknowledgements

Zielsetzung und Themenschwerpunkte (Objectives and Key Themes)

This thesis explores the concept of endotrivial modules over preprojective algebras, focusing on Dynkin types ADE and B, C, F, and G2. The primary objective is to classify these modules, building upon existing results for Dynkin type ADE and extending the analysis to other types.

• Endotrivial modules and their properties
• Reflection functors and their role in module classification
• Preprojective algebras and their relation to Cartan matrices
• Representation theory of modulated graphs
• The role of the Weyl group in module classification

Zusammenfassung der Kapitel (Chapter Summaries)

• Introduction: This chapter provides an overview of the thesis, introducing the concept of endotrivial modules and their significance in representation theory. It highlights the key results of Crawley-Boevey and the connection between endotrivial modules and rigid modules, as well as their relevance to geometrical aspects of representation theory.
• Modules over preprojective algebras: This chapter lays the foundation for the thesis, introducing the concept of preprojective algebras and their associated modules. It delves into the properties of reflection functors and their application in classifying endotrivial modules over Dynkin type ADE algebras.
• Quivers with relations for symmetrizable Cartan matrices: This chapter expands the focus to include symmetrizable Cartan matrices and their associated algebras. It introduces the concept of modulated graphs and explores their representation theory, laying the groundwork for the classification of endotrivial modules for Dynkin types B, C, F, and G2.

Schlüsselwörter (Keywords)

This thesis centers around the concepts of endotrivial modules, preprojective algebras, reflection functors, symmetrizable Cartan matrices, modulated graphs, and representation theory. It explores the classification of endotrivial modules over various Dynkin types, utilizing established results and extending them to new contexts.