Editing Adjoint representation (section)

{{Short description|Mathematical term}}
{{Redirect|Adjoint map|the term in functional analysis|adjoint operator}}
{{Lie groups |Algebras}}
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 map|linear transformations]] of the group's [[Lie algebra]], considered as a [[vector space]]. For example, if ''G'' is <math>\mathrm{GL}(n, \mathbb{R})</math>, the Lie group of real [[invertible matrix|''n''-by-''n'' invertible matrices]], then the adjoint representation is the group homomorphism that sends an invertible ''n''-by-''n'' matrix <math> g </math> to an [[endomorphism]] of the vector space of all linear transformations of <math>\mathbb{R}^n</math> defined by: <math> x \mapsto g x g^{-1} </math>.

For any Lie group, this natural [[group representation|representation]] is obtained by linearizing (i.e. taking the [[Differential of a function|differential]] of) the [[Group action (mathematics)|action]] of ''G'' on itself by [[conjugation (group theory)|conjugation]]. The adjoint representation can be defined for [[linear algebraic group]]s over arbitrary [[field (mathematics)|fields]].