Classical Lie Groups

Abstract

In this chapter, we apply the general theory to classical matrix groups such as $\mathop {\mathrm {GL}}\nolimits _{n}({\mathbb{C}}), \mathop {\rm SL}\nolimits _{n}({\mathbb{C}}), \mathop {\rm SO}\nolimits _{n}({\mathbb{C}}), \mathop {\rm Sp}\nolimits _{2n}({\mathbb{C}})$, and some of their real forms to provide explicit structural and topological information. We will start with compact real forms, i.e., $\mathop {\rm U{}}\nolimits _{n}({\mathbb{K}})$ and $\mathop {\rm SU}\nolimits _{n}({\mathbb{K}})$, where ${\mathbb{K}}$ is ℝ, ℂ, or ℍ, since many results can be reduced to compact groups.