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
Famous quotes containing the words formal and/or definition:
“On every formal visit a child ought to be of the party, by way of provision for discourse.”
—Jane Austen (17751817)
“The very definition of the real becomes: that of which it is possible to give an equivalent reproduction.... The real is not only what can be reproduced, but that which is always already reproduced. The hyperreal.”
—Jean Baudrillard (b. 1929)