In 1870, Sophus Lie was studying the symmetries of differential equations, which generally form “continuous” groups. The analogous problem for polynomials was solved by Galois previously, so there was incentive to solve the related problem for these new kinds of groups. We’ll motivate such continuous groups by looking at matrices, assuming only a little linear algebra.

Lie started a study of simplicity in these continuous groups. Lie understood these groups as groups generated by infinitesimal elements, which led him to believe that a group $G$ should be generalised to consider infinitesimal elements.

Today we separate the infinitesimal elements of a group $G$ to form a Lie algebra $g$, which captures most of the important structures of $G$, but is easier to handle. This talk will focus on motivating Lie's definitions, and provide some techniques used to prove simplicity for Lie groups. As well, some specific examples such as $\mathrm{O}(n)$, $\mathrm{SO}(n)$, $\mathrm{U}(n)$, $\mathrm{SU}(n)$, and $\mathrm{Sp}(n)$ will be mentioned to illustrate the concepts presented.

Please read this basic overview of Lie Theory.