adjoint representation

Every Lie group carries a built-in representation on its own Lie algebra: it acts on its tangent space at the identity by conjugation. This is the adjoint representation. It answers 'how does the group reshuffle its own infinitesimal directions?' — a self-portrait of the group acting on itself.

At the group level, Ad(g) X = g X g^-1: conjugating an algebra element X by a group element g. Because conjugation preserves the Lie algebra and respects the bracket, each Ad(g) is an invertible linear map on g, and Ad : G -> GL(g) is a genuine representation. Differentiating Ad at the identity gives the algebra-level version ad(X) Y = [X, Y]: bracketing IS the infinitesimal adjoint action.

So the bracket reappears as a representation: ad : g -> gl(g), ad(X) = [X, -]. The Jacobi identity is exactly the statement that ad is a Lie algebra homomorphism. These two maps connect through the exponential: Ad(exp X) = exp(ad X), tying the group's conjugation to the algebra's bracket through one clean exponential formula.

Why it matters: the adjoint representation organizes the group's internal structure — its kernel is the center, and for simple groups it is faithful, so the group is essentially recovered from how it acts on its own algebra. A caveat: Ad and ad are easy to confuse — capital Ad is the GROUP acting (conjugation by g), lowercase ad is the ALGEBRA acting (bracket by X); they are linked by Ad(exp X) = exp(ad X).