Compact Groups

Abstract

In this chapter the simply connected groups which are universal coverings of compact groups are studied. It is proved that the Lie algebra $\mathfrak {g}$ of a compact group G decomposes in the direct sum $\mathfrak {g}=\mathfrak {z}\left ( \mathfrak {g} \right ) \oplus \mathfrak {k}$, where $\mathfrak {z}\left ( \mathfrak {g}\right )$ is the center of $\mathfrak {g}$ and $\mathfrak {k}$ is a semi-simple algebra. The simply connected group associated with $\mathfrak {g}$ is the direct product of the simply connected groups of $\mathfrak {z}\left ( \mathfrak {g}\right )$ and $\mathfrak {k}$. The latter is a compact group, a result of the Weyl theorem, which is proved here.