In mathematics, the adjoint representation (or adjoint action) of a Lie groupG is a way of representing the elements of the group as linear transformations of the group's Lie algebra, considered as a vector space. For example, in the case where G is the Lie group of invertible matrices of size n, GL(n), the Lie algebra is the vector space of all (not necessarily invertible) n-by-n matrices. So in this case the adjoint representation is the vector space of n-by-n matrices , and any element g in GL(n) acts as a linear transformation of this vector space given by conjugation: .
For each g in G, define Adg to be the derivative of Ψg at the origin:
where d is the differential and is the tangent space at the origin e (e being the identity element of the group G). Since is a Lie group automorphism, Adg is a Lie algebra automorphism; i.e., an invertible linear transformation of to itself that preserves the Lie bracket. The map
Ad and ad are related through the exponential map: Specifically, Adexp(x) = exp(adx) for all x in the Lie algebra. It is a consequence of the general result relating Lie group and Lie algebra homomorphisms via the exponential map.
The upper-case/lower-case notation is used extensively in the literature. Thus, for example, a vector x in the algebra generates a vector fieldX in the group G. Similarly, the adjoint map adxy = [x,y] of vectors in is homomorphic to the Lie derivativeLXY = [X,Y] of vector fields on the group G considered as a manifold.
Let be a Lie algebra over a field k. Given an element x of a Lie algebra , one defines the adjoint action of x on as the map
for all y in . It is called the adjoint endomorphism or adjoint action. Then there is the linear mapping
given by x ↦ adx. Within End, the Lie bracket is, by definition, given by the commutator of the two operators:
where denotes composition of linear maps. Using the above definition of the Lie bracket, the Jacobi identity
takes the form
where x, y, and z are arbitrary elements of .
This last identity says that ad is a Lie algebra homomorphism; i.e., a linear mapping that takes brackets to brackets. Hence, ad is a representation of a Lie algebra and is called the adjoint representation of the algebra .
If is finite-dimensional, then End is isomorphic to , the Lie algebra of the general linear group over the vector space and if a basis for it is chosen, the composition corresponds to matrix multiplication.
In a more module-theoretic language, the construction simply says that is a module over itself.
The kernel of ad is the center of . Next, we consider the image of ad. Recall that a derivation on a Lie algebra is a linear map that obeys the Leibniz' law, that is,
for all x and y in the algebra.
That adx is a derivation is a consequence of the Jacobi identity. This implies that the image of under ad is
a subalgebra of Der, the space of all derivations of .
When is the Lie algebra of a Lie group G, ad is the differential of Ad at the identity element of G.
If G is abelian of dimension n, the adjoint representation of G is the trivial n-dimensional representation.
If G is a matrix Lie group (i.e. a closed subgroup of GL(n,ℂ)), 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 Adg(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.
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 componentG0 of G. By the first isomorphism theorem we have
Given a finite-dimensional real Lie algebra , by Lie's third theorem, there is a connected Lie group whose Lie algebra is the image of the adjoint representation of (i.e., .) It is called the adjoint group of .
Now, if is the Lie algebra of a connected Lie group G, then is the image of the adjoint representation of G: .
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 torusT. 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.
Let us compute the root system for one of the simplest cases of Lie Groups. Let us consider the group SL(2, R) of two dimensional matrices with determinant 1. This consists of the set of matrices of the form:
with a, b, c, d real and ad − bc = 1.
A maximal compact connected abelian Lie subgroup, or maximal torus T, is given by the subset of all matrices of the form
with . The Lie algebra of the maximal torus is the Cartan subalgebra consisting of the matrices
If we conjugate an element of SL(2, R) by an element of the maximal torus we obtain
are then 'eigenvectors' of the conjugation operation with eigenvalues . The function Λ which gives is a multiplicative character, or homomorphism from the group's torus to the underlying field R. The function λ giving θ is a weight of the Lie Algebra with weight space given by the span of the matrices.
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).