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# Contragredient representation

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 TeG, the tangent space to the identity element in G). Define a map

$Psi : G to mathrm\left\{Aut\right\}\left(G\right),$
by the equation $Psi\left(g\right)= Psi_g$ for all g in G, where $mathrm\left\{Aut\right\}\left(G\right)$ is the automorphism group of G and the automorphism $Psi_g$ is defined by
$Psi_g\left(h\right) = ghg^\left\{-1\right\},$
for all h in G. It follows that the derivative of Ψg at the identity is an automorphism of the Lie algebra $mathfrak g$. We denote this map by Adg:
$mathrm\left\{Ad\right\}_gcolon mathfrak g to mathfrak g.$
To say that Adg is a Lie algebra automorphism is to say that Adg is a linear transformation of $mathfrak g$ that preserves the Lie bracket. The map
$mathrm\left\{Ad\right\}colon G to mathrm\left\{Aut\right\}\left(mathfrak g\right)$
which sends g to Adg is called the adjoint representation of G. This is indeed a representation of G since $mathrm\left\{Aut\right\}\left(mathfrak g\right)$ is a Lie subgroup of $mathrm\left\{GL\right\}\left(mathfrak g\right)$ 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

$mathrm\left\{Ad\right\}colon G to mathrm\left\{Aut\right\}\left(mathfrak g\right)$
gives the adjoint representation of the Lie algebra $mathfrak g$:
$mathrm\left\{ad\right\}colon mathfrak g to mathrm\left\{Der\right\}\left(mathfrak g\right).$
Here $mathrm\left\{Der\right\}\left(mathfrak g\right)$ is the Lie algebra of $mathrm\left\{Aut\right\}\left(mathfrak g\right)$ 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
$mathrm\left\{ad\right\}_x\left(y\right) = \left[x,y\right],$
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 $mathfrak\left\{gl\right\}_n\left(mathbb C\right)$). In this case, the adjoint map is given by Adg(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

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

The image of G under the adjoint representation is denoted by AdG. 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

$mathrm\left\{Ad\right\}_G cong G/C_G\left(G_0\right).$

## 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=SLn(R). We can take the group of diagonal matrices diag(t1,...,tn) as our maximal torus T. Conjugation by an element of T sends

$begin\left\{bmatrix\right\}$
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 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 eiej.

## 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.

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