# Adjoint representation

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, in the case where *G* is the Lie group of invertible matrices of size *n*, *GL(n)*, the Lie algebra is the vector space of all (not necessarily invertible) *n*-by-*n* matrices. So in this case the adjoint representation is the vector space of *n*-by-*n* matrices, and any element *g* in *GL(n)* acts as a linear transformation of this vector space given by conjugation: .

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.

## Formal definition

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Let *G* be a Lie group and let be its Lie algebra (which we identify with *T _{e}G*, the tangent space to the identity element in

*G*). Define a map

by the equation Ψ(*g*) = Ψ_{g} for all *g* in *G*, where Aut(*G*) is the automorphism group of *G* and the automorphism Ψ_{g} is defined by

for all *h* in *G*. It follows that the derivative of Ψ_{g} at the identity is an automorphism of the Lie algebra .

We denote this map by Ad_{g}:

To say that Ad_{g} is a Lie algebra automorphism is to say that Ad_{g} is a linear transformation of that preserves the Lie bracket. The map

which sends *g* to Ad_{g} is called the **adjoint representation** of *G*. This is indeed a representation of *G* since is a Lie subgroup of 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

gives the **adjoint representation** of the Lie algebra :

Here is the Lie algebra of which may be identified with the derivation algebra of . 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