A Quick Introduction to Simple Lie Algebras and Their Representations

Abstract

Simple Lie algebras, their matrix representations and their subalgebra structure have long been studied by many mathematicians and physicists. Since physicists are interested in the Hilbert space of a quantum-mechanical theory, the traditional approach is to form the generators of a symmetry of a Lagrangian from the fields and then to analyze the symmetry directly in terms of the quantum operators. However, this approach is often cumbersome for dealing with the big groups that appear in string theory. It sometimes helps to proceed somewhat differently, similar to that popular in mathematics. There the focus is on the quantum number labels that appear in the state vectors [1].