Tangent space at identity, the Lie bracket [X,Y], and structure constants
Zoom in on a Lie group at the identity and it flattens into a vector space, theLie algebra. One operation, the bracket, remembers how the group twists.
Drag vectors in the tangent plane and watch the curves they cast onto the group: each vector generates its own one-parameter subgroup.
The Lie algebra of a Lie group is its tangent space at the identity element, equipped with the Lie bracket [X,Y]. For matrix groups, the bracket is simply the matrix commutator [X,Y] = XY − YX. The Lie algebra captures the infinitesimal structure of the group — it's a linearization that's much easier to work with than the curved group manifold.
The Lie algebra of SO(3) is the space of 3×3 skew-symmetric matrices, which can be identified with ℝ³. Each vector in this tangent space generates a one-parameter subgroup via the exponential map. Drag vectors in the tangent plane and watch the “shadow curves” (one-parameter subgroups) they cast onto the group manifold.
Follow the loop: flow by εX, εY, −εX, −εY. It does not close. Shrink ε and notice the gap shrinks but keeps its direction, [X,Y].
The Lie bracket measures the failure of two flows to commute. Follow the sequence: flow by εX, then εY, then −εX, then −εY. If the flows commuted, you'd return to the starting point. The gap — proportional to ε²[X,Y] — shows the bracket in action. The gap direction stays constant as ε shrinks, confirming it's a well-defined infinitesimal quantity.
Key insight: The Lie bracket satisfies two fundamental properties: antisymmetry ([X,Y] = −[Y,X]) and the Jacobi identity ([X,[Y,Z]] + [Y,[Z,X]] + [Z,[X,Y]] = 0). These axioms define what it means to be a Lie algebra.
Switch between so(3), su(2), and sl(2,ℝ). Hover over entries to see each bracket computed, and check that the Jacobi identity holds.
Remarkably, the Lie algebra determines the local structure of the group completely. Groups with the same Lie algebra are locally isomorphic (though they may differ globally). The structure constants encode all the bracket relations in a chosen basis.
Given a basis {e₁, e₂, e₃} for a Lie algebra, the structure constants cᵏⱼᵢ are defined by [eⱼ, eᵢ] = Σᵏ cᵏⱼᵢ eᵏ. Explore the structure constants for so(3), su(2), and sl(2,ℝ) — hover to see individual bracket computations, and verify the Jacobi identity is satisfied.
Explore the constellation of simple Lie algebras: four infinite families and five exceptional ones, which twinkle brighter.
One of the crown jewels of mathematics is the classification of simple Lie algebras into four infinite families (Aₙ, Bₙ, Cₙ, Dₙ) and five exceptional cases (G₂, F₄, E₆, E₇, E₈). This constellation map gives a preview — the exceptional algebras, which have no analog in the classical families, twinkle brighter. We'll explore the classification machinery (root systems and Dynkin diagrams) in detail later.
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