Formal Definition
Let G be a Lie group and let be its Lie algebra (which we identify with TeG, the tangent space to the identity element in G). Define a map
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
for all h in G. It follows that the derivative of Ψg at the identity is an automorphism of the Lie algebra .
We denote this map by Adg:
To say that Adg is a Lie algebra automorphism is to say that Adg is a linear transformation of that preserves the Lie bracket. The map
which sends g to Adg 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.
Read more about this topic: Adjoint Representation Of A Lie Group
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