In mathematics, the adjoint representation (or adjoint action) of a Lie group G is a way of representing the elements of the group as linear transformations of the group's Lie algebra, considered as a vector space. For example, if G is ${\displaystyle \mathrm {GL} (n,\mathbb {R} )}$, the Lie group of real n-by-n invertible matrices, then the adjoint representation is the group homomorphism that sends an invertible n-by-n matrix ${\displaystyle g}$ to an endomorphism of the vector space of all linear transformations of ${\displaystyle \mathbb {R} ^{n}}$ defined by: ${\displaystyle x\mapsto gxg^{-1}}$.

For any Lie group, this natural representation is obtained by linearizing (i.e. taking the differential of) the action of G on itself by conjugation. The adjoint representation can be defined for linear algebraic groups over arbitrary fields.

Let G be a Lie group, and let

be the mapping g ↦ Ψ_g, with Aut(G) the automorphism group of G and Ψ_g: G → G given by the inner automorphism (conjugation)

This Ψ is a group homomorphism (it is a Lie group homomorphism if ${\displaystyle G}$ is connected^{[citation needed]}).

For each g in G, define Ad_g to be the derivative of Ψ_g at the origin:

where d is the differential and ${\displaystyle {\mathfrak {g}}=T_{e}G}$ is the tangent space at the origin e (e being the identity element of the group G). Since ${\displaystyle \Psi _{g}}$ is a Lie group automorphism, Ad_g is a Lie algebra automorphism; i.e., an invertible linear transformation of ${\displaystyle {\mathfrak {g}}}$ to itself that preserves the Lie bracket. Moreover, since ${\displaystyle g\mapsto \Psi _{g}}$ is a group homomorphism, ${\displaystyle g\mapsto \operatorname {Ad} _{g}}$ too is a group homomorphism. Hence, the map

is a group representation called the adjoint representation of G.