# 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 ${\displaystyle {\mathfrak {g}}}$ be its Lie algebra (which we identify with TeG, the tangent space to the identity element in G). Define a map

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

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

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

for all h in G. It follows that the derivative of Ψg at the identity is an automorphism of the Lie algebra ${\displaystyle {\mathfrak {g}}}$.

We denote this map by Adg:

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

To say that Adg is a Lie algebra automorphism is to say that Adg is a linear transformation of ${\displaystyle {\mathfrak {g}}}$ that preserves the Lie bracket. The map

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

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

{{#invoke:main|main}}

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

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

gives the adjoint representation of the Lie algebra ${\displaystyle {\mathfrak {g}}}$:

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

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

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