Compact Lie Groups - Key Properties

Compact Lie groups possess remarkable properties that distinguish them from general Lie groups. These properties make compact groups particularly amenable to analysis and central to representation theory.

Theorem

Complete Reducibility (Peter-Weyl Theorem)

Every continuous finite-dimensional representation of a compact Lie group $G$ is completely reducible: it decomposes as a direct sum of irreducible representations. Moreover, the matrix coefficients of irreducibles are dense in $L^2(G)$ .

This is the fundamental result making representation theory of compact groups tractable. Unlike general Lie groups, there are no indecomposable non-irreducible representations.

Theorem

Weyl's Unitary Trick

Every representation $\rho: G \to GL(V)$ of a compact Lie group admits a $G$ -invariant inner product on $V$ , making $\rho(G)$ consist of unitary operators. This allows using tools from functional analysis.

The proof uses averaging over the group with Haar measure: define $\langle v, w\rangle_G = \int_G \langle \rho(g)v, \rho(g)w\rangle_0 \, d\mu(g)$ for any initial inner product $\langle\cdot, \cdot\rangle_0$ .

Remark

Weyl's integraldimension formula: For irreducible representation $V_\lambda$ of compact semisimple $G$ with highest weight $\lambda$ : $\dim V_\lambda = \prod_{\alpha \in \Phi^+} \frac{\langle \lambda + \rho, \alpha\rangle}{\langle \rho, \alpha\rangle}$ where $\Phi^+$ are positive roots and $\rho$ is half-sum of positive roots. This gives explicit dimensions.

Theorem

Maximal Torus Theorem

For compact connected $G$ :

All maximal tori are conjugate
Every element of $G$ is conjugate to an element of any fixed maximal torus $T$
$G = \bigcup_{g \in G} gTg^{-1}$ (every element lies in some maximal torus)

This reduces questions about $G$ to questions about the torus $T$ and the Weyl group $W = N_G(T)/T$ .

Example

For $SU(n)$ , the maximal torus consists of diagonal unitary matrices $\text{diag}(e^{i\theta_1}, \ldots, e^{i\theta_n})$ with $\sum \theta_i = 0 \pmod{2\pi}$ . This has dimension $n-1$ , the rank of $SU(n)$ .

Integration formulas: The Weyl integration formula expresses integrals over $G$ as integrals over the maximal torus: $\int_G f(g) \, dg = \frac{1}{|W|} \int_T f(t) |\Delta(t)|^2 \, dt$ where $\Delta(t) = \prod_{\alpha > 0}(e^{i\alpha(t)/2} - e^{-i\alpha(t)/2})$ is the Weyl denominator.

Theorem

Compact Form Correspondence

Every complex simple Lie algebra $\mathfrak{g}_\mathbb{C}$ has a unique (up to isomorphism) compact real form $\mathfrak{g}_u$ : a real Lie algebra whose complexification is $\mathfrak{g}_\mathbb{C}$ and which is the Lie algebra of a compact Lie group.

For example, $\mathfrak{su}(n)$ is the compact form of $\mathfrak{sl}_n(\mathbb{C})$ . This correspondence allows transferring results between compact groups and complex algebras.

Remark

The Cartan decomposition states that for non-compact semisimple $G$ , there exists a maximal compact subgroup $K$ such that $G/K$ is diffeomorphic to Euclidean space. This reduces topology questions to the compact case.

These properties make compact Lie groups the most thoroughly understood class of Lie groups, serving as anchors for understanding general Lie theory.