The Adjoint Representation

The adjoint representation is how a Lie group naturally acts on its own Lie algebra. For a group element g, the map Ad(g): X ↦ gXg⊃{−1} conjugates algebra elements. This is a linear transformation on the Lie algebra — so Ad gives a homomorphism from the Lie group to GL(g), the invertible linear maps on the algebra.

Pick a group element g ∈ SO(3) and watch the entire Lie algebra (visualized as a grid of arrows) rotate by Ad(g). The smooth, collective motion of all algebra elements reveals how conjugation acts as a rotation of the algebra space itself — for SO(3), Ad(g) is simply rotation by g.

Choose a basis vector X from the Lie algebra. The map ad(X)(Y) = [X,Y] acts on all other algebra elements simultaneously. For so(3), this creates circular flow lines — particles orbit around the X axis at speeds proportional to their distance from it. This is the infinitesimal version of the adjoint rotation.

The infinitesimal version ad(X)(Y) = [X,Y] gives the Lie bracket as a representation. The Killing form B(X,Y) = tr(ad(X) ad(Y)) is a natural inner product on the algebra that detects semisimplicity: a Lie algebra is semisimple if and only if its Killing form is non-degenerate.

The Killing form B(X,Y) = tr(ad(X) · ad(Y)) is a bilinear form on the Lie algebra constructed purely from the bracket. It serves as a natural “inner product” (though it may not be positive definite). For so(3), the Killing form is proportional to the standard dot product. Explore the heatmap of B(eᵢ, eⱼ) across different algebras and see how non-degeneracy detects semisimplicity.

The adjoint orbit of an algebra element X is the set {Ad(g)(X) : g ∈ G}. For SO(3), adjoint orbits are spheres (centered at the origin in ℝ³). For SL(2,ℝ), the orbits are hyperboloids — reflecting the non-compact, indefinite nature of sl(2,ℝ). Compare the two to see how the group's topology shapes its adjoint action.