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:
“Good gentlemen, look fresh and merrily.
Let not our looks put on our purposes,
But bear it as our Roman actors do,
With untired spirits and formal constancy.”
—William Shakespeare (1564–1616)
“According to our social pyramid, all men who feel displaced racially, culturally, and/or because of economic hardships will turn on those whom they feel they can order and humiliate, usually women, children, and animals—just as they have been ordered and humiliated by those privileged few who are in power. However, this definition does not explain why there are privileged men who behave this way toward women.”
—Ana Castillo (b. 1953)