# Adjoint representation of a Lie group

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

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

$\Psi _{g}(h)=ghg^{-1}\,$ 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:

$d(\Psi _{g})_{e}=\mathrm {Ad} _{g}\colon {\mathfrak {g}}\to {\mathfrak {g}}.$ 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

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

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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 ${\mathfrak {g}}$ :

$d(\mathrm {Ad} )_{x}:T_{x}(G)\to T_{Ad(x)}(\mathrm {Aut} ({\mathfrak {g}}))$ $\mathrm {ad} \colon {\mathfrak {g}}\to \mathrm {Der} ({\mathfrak {g}}).$ 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

$\mathrm {ad} _{x}(y)=[x,y]\,$ ${\mathfrak {gl}}_{n}(\mathbf {C} )$