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.KeywordsClassical Matrix GroupsCompact Real FormExceptional IsomorphismNondegenerate Skew-symmetric Bilinear FormMaximal Compact SubgroupThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.