Description

Lie algebras describe the infinitesimal symmetries associated with classical symmetry groups such as $SO(N)$ and $SU(N)$. Though this connection to the associated groups, known as Lie groups, has very interesting geometric meaning, the Lie algebras themselves have a very rich mathematical structure which can be studied independently. Given an algebraic structure such as a group or a Lie algebra, one can try to study the algebraic object using linear algebra, by considering vector spaces on which the algebraic object acts through linear transformations. Such a vector space, along with a description of how the algebraic objects acts on the space, is known as a representation of the algebraic object. In a physical context, representations of Lie algebras appear naturally, as the quantum states of the physical system must form representations of the underlying symmetry algebra of the system.

The semisimple Lie algebras we consider may be classified by their root systems and associated Dynkin diagrams. These root systems may in turn be used to describe lattices – discrete sets of points in \(\mathbb{R}^n\). A particularly interesting class of such lattices are the type II lattices, otherwise known as even unimodular lattices, which arise only in dimensions divisible by 8. We will see that in dimension \(d\) for \(d \le 24\) these type II lattices have been fully classified, focussing in particular on the \(d=8\) and \(d=16\) cases.

The Dynkin diagrams of the finite semisimple Lie algebras. Image by Tomruen. The six roots of the A2 root system, demonstrating the S3 Weyl symmetry. Image by Mathphysman. This picture shows a projection of the E8 root system into the Coxeter plane. Image by Jgmoxness.
On the left we have the finite Dynkin diagrams, which classify the root system of the semisimple Lie algebras. The middle image shows the root system for the \(A_2\) algebra (the six arrows). The Weyl group of this root system is isomorphic to \(D_3\) and the image of the equilateral triangle has been added to this image to demonstrate this. The image on the right shows a projection of the root system of the Lie algebra \(E_8\).

Prerequisites

Algebra II is a prerequisite for this project. Representation Theory III will be beneficial to this project, but it is not a compulsory prerequisite. Students taking this project will also be expected to have a good understanding of linear algebra

Resources