Questions: Lie Group Representations (Introduction)

Question 1

Which of the following is a compact Lie group?

• A. GL_n(ℝ) — the general linear group
• B. SL_2(ℝ) — the special linear group
• C. SO(3) — the rotation group in 3 dimensions
• D. The additive group (ℝ, +)

Answer: (C)

SO(3) is the group of 3×3 orthogonal matrices with determinant 1. It is compact because the orthogonality condition AᵀA = I constrains the entries (they satisfy Σ aᵢⱼ² = 1 for each column), making it a closed bounded subset of ℝ⁹. GL_n(ℝ) is not compact (matrices can have arbitrarily large entries). SL_2(ℝ) is closed but not bounded. (ℝ, +) is not compact.

Question 2

For compact Lie groups, Maschke's theorem generalizes: every continuous finite-dimensional representation is completely reducible. What replaces the averaging sum (1/|G|)Σ_{g∈G}?

Model answer: Integration with respect to the Haar measure: ∫_G f(g) dg. The Haar measure is the unique (up to scale) left-invariant Borel measure on G, and compactness ensures it has finite total mass (normalized to 1).

For finite groups, averaging is (1/|G|)Σ_{g∈G} f(g). For compact Lie groups, the finite sum becomes an integral over the group with respect to Haar measure. The key properties — left invariance (∫f(hg)dg = ∫f(g)dg) and finite total volume — ensure that the averaging trick from Maschke's proof goes through identically. Non-compact groups lack a finite invariant measure, so complete reducibility can fail.

Question 3

The exponential map exp: 𝔤 → G connects the Lie algebra to the Lie group. For matrix groups, exp(X) = Σ_{n=0}^∞ Xⁿ/n!.

• T. True
• F. False

Answer: True

For a matrix Lie group G ⊆ GL_n(ℝ), the Lie algebra 𝔤 consists of matrices X such that exp(tX) ∈ G for all t ∈ ℝ. The exponential map exp: 𝔤 → G defined by the matrix power series is a local diffeomorphism near the identity. It connects infinitesimal (Lie algebra) information to global (Lie group) information. For example, the Lie algebra of SO(3) is the space of 3×3 skew-symmetric matrices, and exp maps them to rotation matrices.

Question 4

The Peter-Weyl theorem for compact Lie groups is the analogue of which result for finite groups?

• A. Lagrange's theorem
• B. The decomposition of the regular representation into irreducibles
• C. The Sylow theorems
• D. The Jordan-Hölder theorem

Answer: (B)

For a finite group G, the regular representation decomposes as ℂ[G] ≅ ⊕ᵢ Vᵢ^{dᵢ}, and the matrix coefficients of irreducible representations form an orthogonal basis of L²(G). The Peter-Weyl theorem generalizes this to compact groups: the matrix coefficients of all irreducible unitary representations form a complete orthonormal system in L²(G, dg). This is the representation-theoretic foundation of harmonic analysis on compact groups.