Structure of compact connected Lie groups

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

On Lie algebras, one has a canonical decomposition

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

Context. The torus factor encodes the abelian part of $G$ (see connected abelian structure ), while $K$ encodes the “noncommutative core.” The finite quotient reflects the possibility of nontrivial center ; passing to the universal covering group eliminates this finite ambiguity.

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 ).