The Lie algebra of the rotation group SO(3) consists of 3×3 skew-symmetric matrices. What is the dimension of this Lie algebra, and what does it represent geometrically?
ADimension 3 — each element represents an infinitesimal rotation about some axis in ℝ³
BDimension 9 — each element is an arbitrary 3×3 matrix
CDimension 6 — each element represents a rotation and a translation
DDimension 1 — each element represents a rotation angle
The space of 3×3 skew-symmetric matrices (Aᵀ = -A) has dimension 3, with basis elements corresponding to infinitesimal rotations about the x, y, and z axes. The Lie bracket [A,B] = AB - BA is the commutator of matrices, and it captures how infinitesimal rotations interact — the fact that [A,B] ≠ 0 in general reflects the non-commutativity of rotations. The exponential map exp : so(3) → SO(3) sends a skew-symmetric matrix to the rotation it generates: exp(tA) is rotation about the axis of A by angle t|A|.
Question 2 True / False
The exponential map of a Lie group is always surjective (every group element is the exponential of some Lie algebra element).
TTrue
FFalse
Answer: False
Surjectivity holds for compact connected Lie groups (like SO(n), SU(n), and any compact group) but fails in general. The classic counterexample is SL(2, ℝ): the matrix diag(-e, -1/e) for e > 0 is in SL(2, ℝ) but is not the exponential of any element of sl(2, ℝ). For connected Lie groups, every element is a product of exponentials (by the Lie group-Lie algebra correspondence), but it may not be a single exponential.
Question 3 Short Answer
A Lie group homomorphism φ : G → H induces a Lie algebra homomorphism dφ_e : 𝔤 → 𝔥 at the identity. What key property does dφ_e preserve?
Think about your answer, then reveal below.
Model answer: The Lie bracket: dφ_e([X, Y]) = [dφ_e(X), dφ_e(Y)] for all X, Y ∈ 𝔤. A Lie algebra homomorphism is a linear map that preserves the bracket. This is because the Lie bracket of left-invariant vector fields is compatible with group homomorphisms — pushing forward by φ commutes with the bracket. The correspondence between Lie group homomorphisms and Lie algebra homomorphisms is one of the central results of Lie theory.
For simply connected Lie groups, the correspondence is a bijection: every Lie algebra homomorphism 𝔤 → 𝔥 lifts uniquely to a Lie group homomorphism G → H. This means the Lie algebra completely determines the simply connected Lie group — a remarkable linearization of a nonlinear object.
Question 4 True / False
Every compact Lie group admits a bi-invariant Riemannian metric — a metric invariant under both left and right multiplication.
TTrue
FFalse
Answer: True
The construction uses averaging (integration over the group with respect to the Haar measure). Start with any left-invariant metric (obtained by choosing an inner product on the Lie algebra and left-translating). Average it over right translations using the Haar measure of the compact group. The result is bi-invariant. For bi-invariant metrics, the Levi-Civita connection has a beautiful formula: ∇_X Y = ½[X,Y] for left-invariant fields. Geodesics through the identity are one-parameter subgroups, and the Riemannian exponential equals the Lie group exponential.