Adjoint representation

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



G
L
(
n
,

R

)


{\displaystyle GL(n,\mathbb {R} )}
, 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



g


{\displaystyle g}
to an endomorphism of the vector space of all linear transformations of





R


n




{\displaystyle \mathbb {R} ^{n}}
defined by:



x
↦
g
x

g

−
1




{\displaystyle x\mapsto 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.

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