Structure of compact connected Lie groups - Imported Knowlpedia Content

Let $G$ be a compact, connected Lie group with Lie algebra $\mathfrak{g}$ .

Theorem (standard structure decomposition). There exist:

a torus $T$ (a compact connected abelian Lie group),
a simply connected compact semisimple Lie group $K$ ,
and a finite central subgroup $F\subset T\times K$ ,

such that

G \cong (T\times K)/F.

On Lie algebras, one has a canonical decomposition

\mathfrak{g} \cong \mathfrak{z}\oplus [\mathfrak{g},\mathfrak{g}],

where $\mathfrak{z}$ is the center of $\mathfrak{g}$ and $[\mathfrak{g},\mathfrak{g}]$ is semisimple (compare Cartan’s semisimplicity criterion).

Remarks

Context. The torus factor encodes the abelian part of $G$ (see connected abelian structure), while $K$ encodes the semisimple part. The finite quotient records an overlap between their centers. The universal cover has the form $\mathbb R^{\dim T}\times K$ ; its kernel over $G$ need not be finite because the torus contributes a lattice.

This decomposition is one conceptual reason compact Lie groups have especially rigid representation theory, via Peter–Weyl and highest-weight methods (see the highest weight theorem).