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. Mathematics is often defined as the study of topics such as quantity, structure, space, and change. ... This article needs a better explanation of technical details or more context regarding applications or importance to make it more accessible to a general audience, or at least to technical readers outside this specialty. ... Representation theory is the branch of mathematics that studies properties of abstract groups via their representations as linear transformations of vector spaces. ... In mathematics, a Lie algebra is an algebraic structure whose main use lies in studying geometric objects such as Lie groups and differentiable manifolds. ... In mathematics, a symmetry group describes all symmetries of objects. ...

Formal definition

Let G be a Lie group and let be its Lie algebra (which we identify with T_eG, the tangent space to the identity element in G). Define a map Ψ : G → Aut(G) by This article needs a better explanation of technical details or more context regarding applications or importance to make it more accessible to a general audience, or at least to technical readers outside this specialty. ... In mathematics, a Lie algebra is an algebraic structure whose main use lies in studying geometric objects such as Lie groups and differentiable manifolds. ... The tangent space of a manifold is a concept which needs to be introduced when generalizing vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector pointing from one to the other. ... In mathematics, an identity element (or neutral element) is a special type of element of a set with respect to a binary operation on that set. ...

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: In mathematics, an automorphism is an isomorphism from a mathematical object to itself. ... In mathematics, the push forward (or pushforward) of a smooth map F : M → N between smooth manifolds at a point p is, in some sense, the best linear approximation of F near p. ...

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 In mathematics, a linear transformation (also called linear map or linear operator) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. ... A lie bracket can refer to: Lie algebra Lie derivative This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the same title. ...

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. Representation theory is the branch of mathematics that studies properties of abstract groups via their representations as linear transformations of vector spaces. ... In mathematics, a subgroup H of a Lie group G is a Lie subgroup if it is also a submanifold of G. According to Cartans theorem, this is equivalent to H being a closed subset in the topological structure of G. Then the Lie algebra h of H is...

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 In mathematics, if φ: G→H is a homomorphism of Lie groups, and g and h are the Lie algebras of G and H respectively, then the induced map φ* on tangent spaces is a homomorphism of Lie algebras, i. ... In mathematics, the push forward (or pushforward) of a smooth map F : M → N between smooth manifolds at a point p is, in some sense, the best linear approximation of F near p. ...

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 In abstract algebra, a derivation on an algebra A over a field k is a linear map D : A → A that satisfies Leibniz law: D(ab) = (Da)b + a(Db). ...

ad_x(y) = [x,y]

for all . For more information see: adjoint representation of a Lie algebra. In mathematics, the adjoint endomorphism or adjoint action is an endomorphism of Lie algebras that plays a fundamental role in the development of the theory of Lie algebras and Lie groups. ...

Examples

• 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 ), 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 SL₂(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.

In mathematics, an abelian group, also called a commutative group, is a group (G, *) such that a * b = b * a for all a and b in G. Abelian groups are named after Niels Henrik Abel. ... In linear algebra, a determinant is a function depending on n that associates a scalar det(A) to every n×n square matrix A. The fundamental geometric meaning of a determinant is as the scale factor for volume when A is regarded as a linear transformation. ... In linear algebra, the trace of an n-by-n square matrix A is defined to be the sum of the elements on the main diagonal (the diagonal from the upper left to the lower right) of A, i. ... In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. ...

Properties

The following table summarizes the properties of the various maps mentioned in the definition

Lie group homomorphism: Ψgh = ΨgΨh	Lie group automorphism: Ψg(ab) = Ψg(a)Ψg(b)
Lie group homomorphism: Adgh = AdgAdh	Lie algebra automorphism: Adg is linear Adg[x,y] = [Adg(x),Adg(y)]
Lie algebra homomorphism: ad is linear ad[x,y] = [adx,ady]	Lie algebra derivation: adx is linear adx[y,z] = [adx(y),z] + [y,adx(z)]

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 In mathematics, the image of an element x in a set X under the function f : X → Y, denoted by f(x), is the unique y in Y that is associated with x. ... Connected and disconnected subspaces of R². The space A at top is connected; the shaded space B at bottom is not. ... In the various branches of mathematics that fall under the heading of abstract algebra, the kernel of a homomorphism measures the degree to which the homomorphism fails to be injective. ... In mathematics, a faithful representation ρ of a group G on a vector space V is a linear representation in which different elements g of G are represented by distinct linear mappings ρ(g). ... In group theory, the centralizer and normalizer of a subset S of a group G are subgroups of G which have a restricted action on the elements of S and S as a whole, respectively. ... In mathematics, the identity component of a topological group G is the connected component C that contains the identity element e. ... In mathematics, the isomorphism theorems are three theorems, applied widely in the realm of universal algebra, stating the existence of certain natural isomorphisms. ...

Roots of a semisimple Lie group

If G is semisimple, the non-zero 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(t₁,...,t_n) as our maximal torus T. Conjugation by an element of T sends In mathematics, the term semisimple is used in a number of related ways, within different subjects. ... Given a set S of complex matrices, each of which is diagonalizable and any two of which commute under multiplication, it is always possible to diagonalize all the elements of S simultaneously. ... In mathematics, a root system is a configuration of vectors in a Euclidean space satisfying certain geometrical properties. ...

Thus, T acts trivially on the diagonal part of the Lie algebra of G and with eigenvectors t_it_j^-1 on the various off-diagonal entries. The roots of G are the weights diag(t₁,...,t_n)→t_it_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. In algebraic geometry, an algebraic group is a group that is an algebraic variety, such that the multiplication and inverse are given by regular functions on the variety. ...

The co-adjoint representation is the contragredient representation of the adjoint representation. A. 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. In mathematics, if G is a group and ρ is a linear representation of it on the vector space V, then the dual representation is defined over the dual vector space as follows: is the transpose of ρ(g−1) for all g in G. Then is also a representation, as may... In mathematics, groups are often used to describe symmetries of objects. ... In mathematics, a symplectic manifold is a smooth manifold equipped with a closed, nondegenerate 2-form. ... In mathematics Representation theory is the name given to the study of standard representations of abstract mathematical structures. ...