Coxeter and Dynkin Diagrams

TL;DR

This paper provides an introductory overview of Coxeter and Dynkin diagrams, highlighting their role in classifying reflection groups, Lie groups, Lie algebras, and related structures, with connections to representation theory and mathematical physics.

Contribution

It offers a comprehensive introduction to Coxeter and Dynkin diagrams, emphasizing their applications across various mathematical and physical theories, suitable for educational purposes.

Findings

Classify finite reflection groups and lattices

Describe connections to Lie groups and algebras

Explain role in representation theory and physics

Abstract

Coxeter and Dynkin diagrams classify a wide variety of structures, most notably finite reflection groups, lattices having such groups as symmetries, compact simple Lie groups and complex simple Lie algebras. The simply laced or "ADE" Dynkin diagrams also classify finite subgroups of SU(2) and quivers with finitely many indecomposable representations. This introductory tour of Coxeter and Dynkin diagrams, based on the column This Week's Finds in Mathematical Physics, is made to accompany a series of lecture videos.