Adjoint representation

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the adjoint representation (or adjoint action) of a <a href="/facts/Lie_group/a9jCQyjF">Lie group</a> G is a way of representing the elements of the group as <a href="/facts/Linear_map/Q1PBKNVV">linear transformations</a> of the group's <a href="/facts/Lie_algebra/n2gAUkNq">Lie algebra</a>, considered as a <a href="/facts/Vector_space/M1pxMLx2">vector space</a>. For example, if G is 
 
 
 
 
 G
 L
 
 (
 n
 ,
 
 R
 
 )
 
 
 {\displaystyle \mathrm {GL} (n,\mathbb {R} )}
 
, the Lie group of real <a href="/facts/Invertible_matrix/fPqXk3V8">n-by-n invertible matrices</a>, then the adjoint representation is the group homomorphism that sends an invertible n-by-n matrix 
 
 
 
 g
 
 
 {\displaystyle g}
 
 to an <a href="/facts/Endomorphism/s7NNlSea">endomorphism</a> 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 <a href="/facts/Group_representation/FsuuDagQ">representation</a> is obtained by linearizing (i.e. taking the <a href="/facts/Differential_of_a_function/337XJVMC">differential</a> of) the <a href="/facts/Group_action_(mathematics)/Vh9VtUUw">action</a> of G on itself by <a href="/facts/Conjugation_(group_theory)/AFuPIFNW">conjugation</a>. The adjoint representation can be defined for <a href="/facts/Linear_algebraic_group/eMLtu5Nh">linear algebraic groups</a> over arbitrary <a href="/facts/Field_(mathematics)/xAjAS4ko">fields</a>.

Adjoint representation open-in-new

Adjoint representation