Adjoint representation

In mathematics, the adjoint representation (or adjoint action) of a Lie group "G" 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, if "G" is formula_1, 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 formula_2 to an endomorphism of the vector space of all linear transformations of formula_3 defined by: formula_4.
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.
Definition.
Let "G" be a Lie group, and let
be the mapping , 
with Aut("G") the automorphism group of "G" and given by the inner automorphism (conjugation)
This Ψ is a Lie group homomorphism.
For each "g" in "G", define to be the derivative of at the origin:
where is the differential and formula_8 is the tangent space at the origin ( being the identity element of the group ). Since formula_9 is a Lie group automorphism, Ad"g" is a Lie algebra automorphism; i.e., an invertible linear transformation of formula_10 to itself that preserves the Lie bracket. Moreover, since formula_11 is a group homomorphism, formula_12 too is a group homomorphism. Hence, the map
is a group representation called the adjoint representation of "G".
If "G" is an immersed Lie subgroup of the general linear group formula_14 (called immersely linear Lie group), then the Lie algebra formula_15 consists of matrices and the exponential map is the matrix exponential formula_16 for matrices "X" with small operator norms. Thus, for "g" in "G" and small "X" in formula_15, taking the derivative of formula_18 at "t" = 0, one gets:
where on the right we have the products of matrices. If formula_20 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 formula_10.
Succinctly, an adjoint representation is an isotropy representation associated to the conjugation action of "G" around the identity element of "G".
Derivative of Ad.
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
at the identity element gives the adjoint representation of the Lie algebra formula_23 of "G":
where formula_25 is the Lie algebra of formula_26 which may be identified with the derivation algebra of formula_10. One can show that
for all formula_29, where the right hand side is given (induced) by the Lie bracket of vector fields. Indeed, recall that, viewing formula_15 as the Lie algebra of left-invariant vector fields on "G", the bracket on formula_10 is given as: for left-invariant vector fields "X", "Y",
where formula_33 denotes the flow generated by "X". As it turns out, formula_34, roughly because both sides satisfy the same ODE defining the flow. That is, formula_35 where formula_36 denotes the right multiplication by formula_37. On the other hand, since formula_38, by chain rule,
as "Y" is left-invariant. Hence,
which is what was needed to show.
Thus, formula_41 coincides with the same one defined in below. Ad and ad are related through the exponential map: Specifically, Adexp("x") = exp(ad"x") 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.
If "G" is an immersely linear Lie group, then the above computation simplifies: indeed, as noted early, formula_42 and thus with formula_43,
Taking the derivative of this at formula_45, we have:
The general case can also be deduced from the linear case: indeed, let formula_47 be an immersely linear Lie group having the same Lie algebra as that of "G". Then the derivative of Ad at the identity element for "G" and that for "G" coincide; hence, without loss of generality, "G" can be assumed to be "G".
The upper-case/lower-case notation is used extensively in the literature. Thus, for example, a vector in the algebra formula_15 generates a vector field in the group . Similarly, the adjoint map of vectors in formula_15 is homomorphic to the Lie derivative of vector fields on the group considered as a manifold.
Further see the derivative of the exponential map.
Adjoint representation of a Lie algebra.
Let formula_15 be a Lie algebra over some field. Given an element of a Lie algebra formula_15, one defines the adjoint action of on formula_15 as the map 
for all in formula_15. It is called the adjoint endomorphism or adjoint action. (formula_55 is also often denoted as formula_56.) Since a bracket is bilinear, this determines the linear mapping
given by . Within Endformula_58, the bracket is, by definition, given by the commutator of the two operators:
where formula_60 denotes composition of linear maps. Using the above definition of the bracket, the Jacobi identity
takes the form 
where , , and are arbitrary elements of formula_15.
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 formula_15.
If formula_15 is finite-dimensional, then Endformula_58 is isomorphic to formula_67, the Lie algebra of the general linear group of the vector space formula_15 and if a basis for it is chosen, the composition corresponds to matrix multiplication.
In a more module-theoretic language, the construction says that formula_15 is a module over itself.
The kernel of ad is the center of formula_15 (that's just rephrasing the definition). On the other hand, for each element in formula_15, the linear mapping formula_72 obeys the Leibniz' law:
for all and in the algebra (the restatement of the Jacobi identity). That is to say, ad"z" is a derivation and the image of formula_15 under ad is 
a subalgebra of Derformula_58, the space of all derivations of formula_15.
When formula_77 is the Lie algebra of a Lie group "G", ad is the differential of Ad at the identity element of "G" (see #Derivative of Ad above).
There is the following formula similar to the Leibniz formula: for scalars formula_78 and Lie algebra elements formula_79,
Structure constants.
The explicit matrix elements of the adjoint representation are given by the structure constants of the algebra. That is, let {ei} be a set of basis vectors for the algebra, with 
Then the matrix elements for 
adei
are given by
Thus, for example, the adjoint representation of su(2) is the defining rep of so(3).
Properties.
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 "G"0 of "G". By the first isomorphism theorem we have
Given a finite-dimensional real Lie algebra formula_15, by Lie's third theorem, there is a connected Lie group formula_86 whose Lie algebra is the image of the adjoint representation of formula_15 (i.e., formula_88.) It is called the adjoint group of formula_15.
Now, if formula_15 is the Lie algebra of a connected Lie group "G", then formula_86 is the image of the adjoint representation of "G": formula_92.
Roots of a semisimple Lie group.
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("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 off-diagonal 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 "ei"−"ej".
Example SL(2, R).
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.
A maximal compact connected abelian Lie subgroup, or maximal torus "T", is given by the subset of all matrices of the form
with formula_96. 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 formula_100. The function Λ which gives formula_101 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).
Variants and analogues.
The adjoint representation can also be defined for algebraic groups over any field.
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.