Adjoint representation
In mathematics, the adjoint representation (or adjoint action) of a Lie group G is the natural representation of G on its own Lie algebra. This representation is the linearized version of the action of G on itself by conjugation.
Contents
Formal definition
Let G be a Lie group and let be its Lie algebra (which we identify with T_{e}G, the tangent space to the identity element in G). Define a map Ψ : G → Aut(G) by
For each g in G, Ψ_{g} is an automorphism of G. It follows that the derivative of Ψ_{g} at the identity is an automorphism of the Lie algebra . We denote this map by Ad_{g}:
To say that Ad_{g} is an Lie algebra automorphism is to say that Ad_{g} is a linear transformation of that preserves the Lie bracket. The map
which sends g to Ad_{g} is called the adjoint representation of G. This is indeed a representation of G since is a Lie subgroup of and the above adjoint map is a Lie group homomorphism. The dimension of the adjoint representation is the same as the dimension of the group G.
Adjoint representation of a Lie algebra
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. Taking the derivative of the adjoint map
gives the adjoint representation of the Lie algebra :
Here is the Lie algebra of which may be identified with the derivation algebra of . The adjoint representation of a Lie algebra is related in a fundamental way to the structure of that algebra. In particular, one can show that
for all . For more information see: adjoint representation of a Lie algebra.
Examples
 If G is abelian of dimension n, the adjoint representation of G is the trivial ndimensional representation.
 If G is a matrix Lie group (i.e. a closed subgroup of ), then its Lie algebra is an algebra of n×n matrices with the commutator for a Lie bracket (i.e. a subalgebra of ). In this case, the adjoint map is given by Ad_{g}(x) = gxg^{−1}.
 If G is SL_{2}(R) (real 2×2 matrices with determinant 1), the Lie algebra of G consists of real 2×2 matrices with trace 0. The representation is equivalent to that given by the action of G by linear substitution on the space of binary (i.e., 2 variable) quadratic forms.
Properties
The following table summarizes the properties of the various maps mentioned in the definition
Lie group homomorphism:

Lie group automorphism:

Lie group homomorphism:

Lie algebra automorphism:

Lie algebra homomorphism:

Lie algebra derivation:

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 G_{0} of G. By the first isomorphism theorem we have
Roots of a semisimple Lie group
If G is semisimple, the nonzero weights of the adjoint representation form a root system. To see how this works, consider the case G=SL_{n}(R). We can take the group of diagonal matrices diag(t_{1},...,t_{n}) 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 t_{i}t_{j}^{1} on the various offdiagonal entries. The roots of G are the weights diag(t_{1},...,t_{n})→t_{i}t_{j}^{1}. This accounts for the standard description of the root system of G=SL_{n}(R) as the set of vectors of the form e_{i}e_{j}.
Variants and analogues
The adjoint representation can also be defined for algebraic groups over any field.
The coadjoint representation is the contragredient representation of the adjoint representation. A. Kirillov observed that the orbit of any vector in a coadjoint representation is a symplectic manifold. According to the philosophy in representation theory known as the orbit method, the irreducible representations of a Lie group G should be indexed in some way by its coadjoint orbits. This relationship is closest in the case of nilpotent Lie groups.