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Concept
Adjoint representation
Summary
In mathematics, the adjoint representation (or adjoint action) of a Lie group G is a way of representing the elements of the group as linear transformations of the group's Lie algebra, considered as a vector space. For example, if G is , the Lie group of real n-by-n invertible matrices, then the adjoint representation is the group homomorphism that sends an invertible n-by-n matrix to an endomorphism of the vector space of all linear transformations of defined by: .
For any Lie group, this natural representation is obtained by linearizing (i.e. taking the differential of) the action of G on itself by conjugation. The adjoint representation can be defined for linear algebraic groups over arbitrary fields.
Representation theory and Lie group#The Lie algebra associated with a Lie group
Let G be a Lie group, and let
be the mapping g ↦ Ψg,
with Aut(G) the automorphism group of G and Ψg: G → G given by the inner automorphism (conjugation)
This Ψ is a Lie group homomorphism.
For each g in G, define Adg to be the derivative of Ψg at the origin:
where d is the differential and is the tangent space at the origin e (e being the identity element of the group G). Since is a Lie group automorphism, Adg is a Lie algebra automorphism; i.e., an invertible linear transformation of to itself that preserves the Lie bracket. Moreover, since is a group homomorphism, too is a group homomorphism. Hence, the map
is a group representation called the adjoint representation of G.
If G is an immersed Lie subgroup of the general linear group (called immersely linear Lie group), then the Lie algebra consists of matrices and the exponential map is the matrix exponential for matrices X with small operator norms. Thus, for g in G and small X in , taking the derivative of at t = 0, one gets:
where on the right we have the products of matrices. If is a closed subgroup (that is, G is a matrix Lie group), then this formula is valid for all g in G and all X in .
Succinctly, an adjoint representation is an isotropy representation associated to the conjugation action of G around the identity element of G.
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