Adjoint Representation of A Lie Group - Formal Definition

Formal Definition

Let G be a Lie group and let be its Lie algebra (which we identify with T_eG, 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.