Classification of Lie groups

The classification of Lie groups and Lie algebras is a topic that is helpful, but not required, in understanding most of theoretical physics. Nevertheless, many of the concepts and terminology from this area are frequently seen in the physics literature, and so are covered here.

We have already learned two important facts with regard to the classification of Lie groups; namely, that every Lie group has connected components diffeomorphic to the normal subgroup of the identity component, and every connected Lie group \({G}\) has a simply connected universal covering group \({G^{*}}\). \({G}\) is then obtained from \({G^{*}}\) by taking the quotient \({G^{*}/N}\), where \({N}\) is a discrete normal subgroup that is isomorphic to \({\pi_{1}\left(G\right)}\). This discrete subgroup lies in the center of \({G^{*}}\). We then have a picture of how general Lie groups are related to simply connected Lie groups, which are one-to-one with Lie algebras.


Above we see that the identity component \({G^{e}}\) is the quotient of the universal covering group \({G^{*}}\) by a discrete normal subgroup \({N}\). A general Lie group \({G}\) has \({G^{e}}\) as a normal Lie subgroup.

\({G/G^{e}}\) is called the component group of \({G}\), and is not in general a subgroup of \({G}\), so we cannot express \({G}\) as a semidirect product of \({G^{e}}\) and \({G/G^{e}}\). Similarly, since \({G^{e}}\) is not in general a subgroup of \({G^{*}}\), we cannot express \({G^{*}}\) as a semidirect product of \({N}\) and \({G^{e}}\). Fortunately, most Lie groups in physics have at most a small number of connected components, so a classification of connected Lie groups will have a large impact. Unfortunately, classifying connected Lie groups is a vast and complicated field of study. Fortunately, in physics we can profitably narrow our focus to compact connected Lie groups, and still obtain important results. Unfortunately, even with this reduced scope, the subject is quite complicated.

An Illustrated Handbook