7.2 Compact Lie groups and their representations

Compact Lie groups are a special class of Lie groups with unique properties. They have finite measures, discrete centers, and compact derived subgroups. Examples include U(1), SU(n), SO(n), and Sp(n).

Representations of compact Lie groups are continuous homomorphisms to linear groups. They're finite-dimensional and can be decomposed into irreducible parts. The Peter-Weyl theorem provides a powerful tool for understanding these representations and their characters.

Compact Lie groups

Definition and properties

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• A compact Lie group is a Lie group that is compact as a topological space
  • Means it is closed and bounded in the topology induced by any Riemannian metric on the group
• Compact Lie groups have a unique bi-invariant Haar measure
  • This measure is finite and allows integration over the group
• Compact Lie groups have a discrete center and a compact derived subgroup
  • The derived subgroup of a compact Lie group is always closed

Examples

• The circle group U(1) is a compact Lie group
• The special unitary group SU(n) is a compact Lie group
• The special orthogonal group SO(n) for $n \geq 3$ is a compact Lie group
• The quaternionic unitary group Sp(n) is a compact Lie group
• Every compact Lie group has a faithful finite-dimensional representation
  • This means it can be viewed as a subgroup of $GL(n,\mathbb{C})$ for some $n$

Representations of compact Lie groups

Definition and properties

• A representation of a compact Lie group $G$ is a continuous homomorphism $\rho: G \to GL(V)$, where $V$ is a finite-dimensional vector space over $\mathbb{C}$
• Irreducible representations of a compact Lie group are finite-dimensional
  • Every representation can be decomposed into a direct sum of irreducible representations
• The character of a representation $\rho: G \to GL(V)$ is the function $\chi_\rho: G \to \mathbb{C}$ defined by $\chi_\rho(g) = \text{tr}(\rho(g))$
  • Characters are constant on conjugacy classes and determine the representation up to isomorphism
• The regular representation of a compact Lie group $G$ on $L^2(G)$ decomposes into a direct sum of all irreducible representations, each occurring with multiplicity equal to its dimension

Construction using the exponential map

• Representations can be constructed using the exponential map
  • For a compact Lie group $G$ with Lie algebra $\mathfrak{g}$, the exponential map $\exp: \mathfrak{g} \to G$ is surjective
  • This allows representations of $G$ to be constructed from representations of $\mathfrak{g}$
• Example: The fundamental representation of SU(2) can be constructed using the exponential map and the Pauli matrices as a basis for the Lie algebra $\mathfrak{su}(2)$

Peter-Weyl theorem for decomposition

Statement and consequences

• The Peter-Weyl theorem states that the matrix coefficients of irreducible representations of a compact Lie group $G$ form an orthonormal basis for $L^2(G)$
• As a consequence, any representation of $G$ can be decomposed into a direct sum of irreducible representations
• The Peter-Weyl theorem implies that any continuous function on $G$ can be uniformly approximated by linear combinations of matrix coefficients of irreducible representations
  • This is analogous to the Fourier series expansion of periodic functions

Multiplicity and character inner product

• The multiplicity of an irreducible representation in a given representation can be computed using the inner product of characters
  • If $\rho$ is a representation with character $\chi_\rho$ and $\tau$ is an irreducible representation with character $\chi_\tau$, then the multiplicity of $\tau$ in $\rho$ is given by the inner product $\langle \chi_\rho, \chi_\tau \rangle = \int_G \chi_\rho(g) \chi_\tau(g)^* dg$, where $dg$ is the Haar measure
• Example: The multiplicity of the trivial representation in the regular representation of a compact Lie group is equal to 1, since the character of the trivial representation is the constant function 1

Representations vs maximal tori

Maximal tori and the Weyl group

• A torus in a Lie group $G$ is a connected abelian subgroup
  • A maximal torus is a torus that is not properly contained in any other torus
• All maximal tori in a compact Lie group are conjugate
  • Their dimension is called the rank of the group
• The Weyl group $W(G,T) = N_G(T)/T$ acts on the weights of a representation
  • The character of a representation is determined by its weights and their multiplicities

Weight space decomposition

• Representations of a compact Lie group $G$ restrict to representations of a maximal torus $T$
  • These restricted representations decompose into one-dimensional representations, or characters, of $T$
• The weights of a representation $\rho: G \to GL(V)$ are the characters of $T$ that appear in the restriction of $\rho$ to $T$
  • The multiplicity of a weight is the dimension of the corresponding $T$-eigenspace in $V$
• Example: The fundamental representation of SU(2) decomposes into two one-dimensional weight spaces with weights $\pm 1$ when restricted to the maximal torus $T = \{\text{diag}(e^{i\theta}, e^{-i\theta}) : \theta \in \mathbb{R}\}$

Weyl character formula

• The Weyl character formula expresses the character of an irreducible representation in terms of its highest weight and the action of the Weyl group
  • It provides a way to compute characters without explicitly constructing the representation
• Example: For the fundamental representation of SU(2) with highest weight $\lambda = 1$, the Weyl character formula gives $\chi_\lambda(e^{i\theta}) = e^{i\theta} + e^{-i\theta} = 2\cos(\theta)$