Adjoint representation of a Lie group

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Template:Lie groups

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 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.

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

for all .