# Adjoint representation

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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 Ψ : G → Aut(G) by

$\Psi _{g}(h)=ghg^{-1}.\,$ For each g in G, Ψg is an automorphism of 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:

$\mathrm {Ad} _{g}\colon {\mathfrak {g}}\to {\mathfrak {g}}.$ To say that Adg is an Lie algebra automorphism is to say that Adg 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 Adg 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 $\mathrm {GL} _{n}(\mathbb {C} )$ ), 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 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

 $\Psi \colon G\to \mathrm {Aut} (G)\,$ $\Psi _{g}\colon G\to G\,$ Lie group homomorphism: $\Psi _{gh}=\Psi _{g}\Psi _{h}$ Lie group automorphism: $\Psi _{g}(ab)=\Psi _{g}(a)\Psi _{g}(b)$ $(\Psi _{g})^{-1}=\Psi _{g^{-1}}$ $\mathrm {Ad} \colon G\to \mathrm {Aut} ({\mathfrak {g}})$ $\mathrm {Ad} _{g}\colon {\mathfrak {g}}\to {\mathfrak {g}}$ Lie group homomorphism: $\mathrm {Ad} _{gh}=\mathrm {Ad} _{g}\mathrm {Ad} _{h}$ Lie algebra automorphism: $\mathrm {Ad} _{g}$ is linear $(\mathrm {Ad} _{g})^{-1}=\mathrm {Ad} _{g^{-1}}$ $\mathrm {Ad} _{g}[x,y]=[\mathrm {Ad} _{g}(x),\mathrm {Ad} _{g}(y)]$ $\mathrm {ad} \colon {\mathfrak {g}}\to \mathrm {Der} ({\mathfrak {g}})$ $\mathrm {ad} _{x}\colon {\mathfrak {g}}\to {\mathfrak {g}}$ 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 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

$\mathrm {Ad} _{G}\cong G/C_{G}(G_{0}).$ ## 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=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

${\begin{bmatrix}a_{11}&a_{12}&\cdots &a_{1n}\\a_{21}&a_{22}&\cdots &a_{2n}\\\vdots &\vdots &\ddots &\vdots \\a_{n1}&a_{n2}&\cdots &a_{nn}\\\end{bmatrix}}\mapsto {\begin{bmatrix}a_{11}&t_{1}t_{2}^{-1}a_{12}&\cdots &t_{1}t_{n}^{-1}a_{1n}\\t_{2}t_{1}^{-1}a_{21}&a_{22}&\cdots &t_{2}t_{n}^{-1}a_{2n}\\\vdots &\vdots &\ddots &\vdots \\t_{n}t_{1}^{-1}a_{n1}&t_{n}t_{2}^{-1}a_{n2}&\cdots &a_{nn}\\\end{bmatrix}}.$ 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 ei-ej.

## 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. 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, 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.