Adjoint representation of a Lie algebra

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 GL(n) (the Lie group of n-by-n invertible matrices), its Lie algebra is the vector space of all (not necessarily invertible) n-by-n matrices. So in this example, the adjoint representation is the vector space of n-by-n matrices x {displaystyle x} , and any element g in GL(n) acts as a linear transformation of this vector space given by conjugation: x ↦ g x g − 1 {displaystyle xmapsto gxg^{-1}} . 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 GL(n) (the Lie group of n-by-n invertible matrices), its Lie algebra is the vector space of all (not necessarily invertible) n-by-n matrices. So in this example, the adjoint representation is the vector space of n-by-n matrices x {displaystyle x} , and any element g in GL(n) acts as a linear transformation of this vector space given by conjugation: x ↦ g x g − 1 {displaystyle xmapsto gxg^{-1}} . 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. 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 g = T e G {displaystyle {mathfrak {g}}=T_{e}G} is the tangent space at the origin e (e being the identity element of the group G). Since Ψ g {displaystyle Psi _{g}} is a Lie group automorphism, Adg is a Lie algebra automorphism; i.e., an invertible linear transformation of g {displaystyle {mathfrak {g}}} to itself that preserves the Lie bracket. Moreover, since g ↦ Ψ g {displaystyle gmapsto Psi _{g}} is a group homomorphism, g ↦ Ad g {displaystyle gmapsto operatorname {Ad} _{g}} 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 G L n ( C ) {displaystyle mathrm {GL} _{n}(mathbb {C} )} (called immersely linear Lie group), then the Lie algebra g {displaystyle {mathfrak {g}}} consists of matrices and the exponential map is the matrix exponential exp ⁡ ( X ) = e X {displaystyle operatorname {exp} (X)=e^{X}} for matrices X with small operator norms. Thus, for g in G and small X in g {displaystyle {mathfrak {g}}} , taking the derivative of Ψ g ( exp ⁡ ( t X ) ) = g e t X g − 1 {displaystyle Psi _{g}(operatorname {exp} (tX))=ge^{tX}g^{-1}} at t = 0, one gets: