, the Lie group of real n-by-n invertible matrices, then the adjoint representation is the group homomorphism that sends an invertible n-by-n matrix
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.
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).
is a closed subgroup (that is, G is a matrix Lie group), then this formula is valid for all g in G and all X in
, where the right hand side is given (induced) by the Lie bracket of vector fields.
as the Lie algebra of left-invariant vector fields on G, the bracket on
, roughly because both sides satisfy the same ODE defining the flow.
[5] If G is an immersely linear Lie group, then the above computation simplifies: indeed, as noted early,
is homomorphic[clarification needed] to the Lie derivative LXY = [X,Y] of vector fields on the group G considered as a manifold.
Since a bracket is bilinear, this determines the linear mapping given by x ↦ adx.
Using the above definition of the bracket, the Jacobi identity takes the form where x, y, and z are arbitrary elements of
is the Lie algebra of square matrices and the composition corresponds to matrix multiplication.
obeys the Leibniz' law: for all x and y in the algebra (the restatement of the Jacobi identity).
is the Lie algebra of a Lie group G, ad is the differential of Ad at the identity element of G. There is the following formula similar to the Leibniz formula: for scalars
, The explicit matrix elements of the adjoint representation are given by the structure constants of the algebra.
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 Given a finite-dimensional real Lie algebra
If G is semisimple, the non-zero weights of the adjoint representation form a root system.
[6] (In general, one needs to pass to the complexification of the Lie algebra before proceeding.)
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.
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: with a, b, c, d real and ad − bc = 1.
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 The matrices are then 'eigenvectors' of the conjugation operation with eigenvalues
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.
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.