For any Lie group, this natural representation is obtained by linearizing (i.e. taking the differential of) the action of G on itself by conjugation. The adjoint representation can be defined for linear algebraic groups over arbitrary fields.
Succinctly, an adjoint representation is an isotropy representation associated to the conjugation action of G around the identity element of G.
One may always pass from a representation of a Lie group G to a representation of its Lie algebra by taking the derivative at the identity.
Thus, for example, the adjoint representation of su(2) is the defining rep of so(3).
The following table summarizes the properties of the various maps mentioned in the definition
The image of G under the adjoint representation is denoted by Ad(G). If G is connected, the kernel of the adjoint representation coincides with the kernel of Ψ which is just the center of G. Therefore, the adjoint representation of a connected Lie group G is faithful if and only if G is centerless. More generally, if G is not connected, then the kernel of the adjoint map is the centralizer of the identity component G0 of G. By the first isomorphism theorem we have
If G is semisimple, the non-zero weights of the adjoint representation form a root system. (In general, one needs to pass to the complexification of the Lie algebra before proceeding.) To see how this works, consider the case G = SL(n, R). We can take the group of diagonal matrices diag(t1, ..., tn) as our maximal torus T. Conjugation by an element of T sends
Thus, T acts trivially on the diagonal part of the Lie algebra of G and with eigenvectors titj−1 on the various off-diagonal entries. The roots of G are the weights diag(t1, ..., tn) → titj−1. This accounts for the standard description of the root system of G = SLn(R) as the set of vectors of the form ei−ej.
When computing the root system for one of the simplest cases of Lie Groups, the group SL(2, R) of two dimensional matrices with determinant 1 consists of the set of matrices of the form:
A maximal compact connected abelian Lie subgroup, or maximal torus T, is given by the subset of all matrices of the form
If we conjugate an element of SL(2, R) by an element of the maximal torus we obtain
It is satisfying to show the multiplicativity of the character and the linearity of the weight. It can further be proved that the differential of Λ can be used to create a weight. It is also educational to consider the case of SL(3, R).
The co-adjoint representation is the contragredient representation of the adjoint representation. Alexandre Kirillov observed that the orbit of any vector in a co-adjoint representation is a symplectic manifold. According to the philosophy in representation theory known as the orbit method (see also the Kirillov character formula), the irreducible representations of a Lie group G should be indexed in some way by its co-adjoint orbits. This relationship is closest in the case of nilpotent Lie groups.