Chapter IV. Compact Lie Groups

Abstract

This chapter is about structure theory for compact Lie groups, and a certain amount of representation theory is needed for the development. The first section gives examples of group representations and shows how to construct new representations from old ones by using tensor products and the symmetric and exterior algebras.

In the abstract representation theory for compact groups, the basic result is Schur's Lemma, from which the Schur orthogonality relations follow. A deeper result is the Peter-Weyl Theorem, which guarantees a rich supply of irreducible representations. From the Peter-Weyl Theorem it follows that any compact Lie group can be realized as a group of real or complex matrices.

The Lie algebra of a compact Lie group admits an invariant inner product, and consequently such a Lie algebra is reductive. From Chapter I it is known that a reductive Lie algebra is always the direct sum of its center and its commutator subalgebra. In the case of the Lie algebra of a compact connected Lie group, the analytic subgroups corresponding to the center and the commutator subalgebra are closed. Consequently the structure theory of compact connected Lie groups in many respects reduces to the semisimple case.

If $T$ is a maximal torus of a compact connected Lie group $G$, then each element of $G$ is conjugate to a member of $T$. It follows that the exponential map for $G$ is onto and that the centralizer of a torus is always connected. The analytically defined Weyl group $W(G,T)$ is the quotient of the normalizer of $T$ by the centralizer of $T$, and it coincides with the Weyl group of the underlying root system.

Weyl's Theorem says that the fundamental group of a compact semisimple Lie group $G$ is finite. Hence the universal covering group of $G$ is compact.