the adjoint representation

A group acts on itself by conjugation: g sends x to g x g^(-1), the 'change of viewpoint' that re-expresses a symmetry in rotated coordinates. Since conjugation fixes the identity, differentiating it gives an action of the group on the tangent space at the identity — that is, on the Lie algebra. This is the adjoint representation: it is how the group sees its own infinitesimal symmetries, and it is the single most important representation a Lie group carries because it is built from nothing but the group itself.

There are two layered pieces, written Ad (capital) and ad (lower case). For each g in G, conjugation C_g(x) = g x g^(-1) fixes e, so its differential at e is a linear map Ad(g) = (dC_g)_e : g -> g. The assignment Ad: G -> GL(g) is a representation of G on its Lie algebra: Ad(g h) = Ad(g) Ad(h). On a matrix group Ad(g) X = g X g^(-1), ordinary matrix conjugation. Differentiating Ad itself at the identity gives ad: g -> gl(g), the adjoint representation of the Lie algebra, and the punchline is that ad is just the bracket: ad(X)(Y) = [X, Y]. So the bracket IS the infinitesimal conjugation. The two fit together by exp(Ad(g)X) = g exp(X) g^(-1) and Ad(exp(X)) = e^(ad(X)) = I + ad(X) + ad(X)^2/2! + ... .

The adjoint representation is the backbone of structure theory: the Killing form is built from ad, the kernel of Ad is the center of G (for connected G), the image Ad(G) is the inner automorphisms, and a group is semisimple precisely when its adjoint representation is well-behaved in a sense made precise by the Killing form. A frequent confusion to avoid: Ad is a representation of the GROUP on the vector space g, while ad is a representation of the ALGEBRA on g; they are linked by differentiation but live at different levels. Also Ad need not be faithful — its kernel is exactly the center, so for an abelian group Ad is trivial.