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.

Formal definition

Let G be a Lie group and let $mathfrak\; g$ be its Lie algebra (which we identify with T_eG, the tangent space to the identity element in G). Define a map

$Psi\; :\; G\; to\; mathrm\{Aut\}(G),$

by the equation $Psi(g)=\; Psi\_g$ for all g in G, where $mathrm\{Aut\}(G)$ is the automorphism group of G and the automorphism $Psi\_g$ is defined by

$Psi\_g(h)\; =\; ghg^\{-1\},$

for all h in G. It follows that the derivative of Ψ_g at the identity is an automorphism of the Lie algebra $mathfrak\; g$. We denote this map by Ad_g:

$mathrm\{Ad\}\_gcolon\; mathfrak\; g\; to\; mathfrak\; g.$

To say that Ad_g is a Lie algebra automorphism is to say that Ad_g is a linear transformation of $mathfrak\; g$ that preserves the Lie bracket. The map

$mathrm\{Ad\}colon\; G\; to\; mathrm\{Aut\}(mathfrak\; g)$

which sends g to Ad_g is called the adjoint representation of G. This is indeed a representation of G since $mathrm\{Aut\}(mathfrak\; g)$ is a Lie subgroup of $mathrm\{GL\}(mathfrak\; g)$ 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

$mathrm\{Ad\}colon\; G\; to\; mathrm\{Aut\}(mathfrak\; g)$

gives the adjoint representation of the Lie algebra $mathfrak\; g$:

$mathrm\{ad\}colon\; mathfrak\; g\; to\; mathrm\{Der\}(mathfrak\; g).$

Here $mathrm\{Der\}(mathfrak\; g)$ is the Lie algebra of $mathrm\{Aut\}(mathfrak\; g)$ which may be identified with the derivation algebra of $mathfrak\; g$. 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

$mathrm\{ad\}\_x(y)\; =\; [x,y],$

for all $x,y\; in\; mathfrak\; g$. 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 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 $mathfrak\{gl\}\_n(mathbb\; C)$). 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.

Properties

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

$Psicolon\; G\; to\; mathrm\{Aut\}(G),$	$Psi\_gcolon\; G\; to\; G,$
Lie group homomorphism: $Psi\_\{gh\}\; =\; Psi\_gPsi\_h$	Lie group automorphism: $Psi\_g(ab)\; =\; Psi\_g(a)Psi\_g(b)$ $(Psi\_g)^\{-1\}\; =\; Psi\_\{g^\{-1\}\}$
$mathrm\{Ad\}colon\; G\; to\; mathrm\{Aut\}(mathfrak\; g)$	$mathrm\{Ad\}\_gcolon\; mathfrak\; g\; to\; mathfrak\; g$
Lie group homomorphism: $mathrm\{Ad\}\_\{gh\}\; =\; mathrm\{Ad\}\_gmathrm\{Ad\}\_h$	Lie algebra automorphism: $mathrm\{Ad\}\_g$ is linear $(mathrm\{Ad\}\_g)^\{-1\}\; =\; mathrm\{Ad\}\_\{g^\{-1\}\}$ $mathrm\{Ad\}\_g[x,y]\; =\; [mathrm\{Ad\}\_g(x),mathrm\{Ad\}\_g(y)]$
$mathrm\{ad\}colon\; mathfrak\; g\; to\; mathrm\{Der\}(mathfrak\; g)$	$mathrm\{ad\}\_xcolon\; mathfrak\; g\; to\; mathfrak\; g$
Lie algebra homomorphism: $mathrm\{ad\}$ is linear $mathrm\{ad\}\_\{[x,y]\}\; =\; [mathrm\{ad\}\_x,mathrm\{ad\}\_y]$	Lie algebra derivation: $mathrm\{ad\}\_x$ is linear $mathrm\{ad\}\_x[y,z]\; =\; [mathrm\{ad\}\_x(y),z]\; +\; [y,mathrm\{ad\}\_x(z)]$

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

$mathrm\{Ad\}\_G\; cong\; G/C\_G(G\_0).$

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

$begin\{bmatrix\}$

a_{11}&a_{12}&cdots&a_{1n} a_{21}&a_{22}&cdots&a_{2n} vdots&vdots&ddots&vdots a_{n1}&a_{n2}&cdots&a_{nn} end{bmatrix} mapsto begin{bmatrix} a_{11}&t_1t_2^{-1}a_{12}&cdots&t_1t_n^{-1}a_{1n} t_2t_1^{-1}a_{21}&a_{22}&cdots&t_2t_n^{-1}a_{2n} vdots&vdots&ddots&vdots t_nt_1^{-1}a_{n1}&t_nt_2^{-1}a_{n2}&cdots&a_{nn} end{bmatrix}.

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.

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.

References