Lie algebras and reflection groups

In this module, we will introduce the concept of Lie algebras. These are a special type of algebras (i.e. vector spaces with a compatible ring structure) whose products are anti-commutative and satisfy the Jacobi identity.

Lie algebras were first introduced by Marius Sophus Lie in the 1870s (and independently by Wilhelm Killing in 1880s) in the study of differential equations with symmetries. They arose from considering infinitessimally small actions of (smooth) symmetry groups on these systems. Analogously, imposing symmetry conditions on other problems naturally lead to applications of the theory of Lie algebras in these areas. Moreover, Lie algebras arise naturally in many contexts, such as differential geometry and quantum mechanics.

Another large part of their significance is the ‘‘classifiability’’ of the theory: finite-dimensional semisimple Lie algebras can be decomposed into simple parts that cannot be divided further (similar to ‘‘irreducibility’’ in representation theory), and these simple parts belong to a finite number of families. This in turn allows applications to be reduced to more tractable cases and solved there.

An important role in the classification of simple Lie algebras is played by reflection groups. These groups generalize the well-known examples of finite groups such as the dihedral groups: a group generated by reflections. Finite reflection groups are determined by their root systems and turn out to allow a description as finite Coxeter groups. These have a succinct description in terms of Coxeter graphs that allow an easy classification. We then discuss the special case of Weyl groups whose root systems are crystallographic and determine the classification of simple Lie algebras.

This course will consist of three parts, following two books by J.E. Humphreys, namely Introduction to Lie Algebras and Representation Theory and Reflection groups and Coxeter groups. Purchasing these books is not required, there will be lecture notes available on the OPAL page.

In the first, which will most likely consist of the first half of the course, we will start with the basics of Lie algebras. Specifically, we will work through Chapters I and II of the Lie algebras book. Topics will include:

• Fundamentals and examples of Lie algebras and their ideals.
• Solvability, nilpotency and semisimplicity of Lie algebras.
• Theorems of Engel and Lie and Cartan's Criterion.
• The Jordan-Chevalley decomposition and the Killing form.
• Representations of semisimple Lie algebras and their complete reducibility.
• Roots of the adjoint representation of a semisimple Lie algebra.
• Root space decompositions of semisimple Lie algebras and weight space decompositions of representations.

In the second part, we will move to (finite) reflection groups, Coxeter groups, Weyl groups and their respective classifications. This part replace part III of the Lie Algebras book with the more general general discussion in the Reflection groups textbook. Specifically we will work through Sections 1 and 2 of Chapter I, which will approximately take up another quarter of the course. We will discuss:

• Definition of reflection groups and the abstract theory of root systems.
• Positive and simple systems, and their existence, conjugacy and uniqueness.
• Simple reflections and reduced expressions.
• Coxeter groups and Coxeter graphs.
• Parabolic subgroups and irreducible components.
• Classification of Coxeter graphs.
• Crystallographic root systems, Weyl groups and Dynkin diagrams.

In the third and final part, spanning the final quarter of the course, we will use the Dynkin diagrams of the roots of the adjoint representation to classify semisimple Lie algebras. This part will return to the Lie algebras textbook, namely Chapters IV and and V. We will most likely not have sufficient time to go into full detail of the proof, but we will introduce the most important concepts and try to sketch the arguments of the existence and uniqueness statements for semisimple Lie algebras. In particular, we will see:

• The decomposition of a semisimple Lie algebra in its simple parts as determined by its Dynkin diagram.
• Uniqueness of a simple Lie algebra with a given Dynkin diagram.
• Engel, Cartan and Borel subalgebras, and their conjugacy.
• Universal enveloping algebras and the Poincaré-Birkhoff-Witt Theorem.
• Serre's Theorem and the existence of a semisimple Lie algebra for any given Dynkin diagram.

Prerequisites for the module: The theory of Lie algebras is mostly self-contained, however a good understanding of linear algebra is required. Basic knowledge of algebra will be helpful for the course.
Any knowledge of differential systems or physics is not required, but might help motivate the theory.