Adjoint Lie algebra representation
The adjoint representation of a Lie algebra
The Jacobi identity implies
Properties
a d π β‘ [ π , π ] = [ a d π β‘ π , π ] + [ π , a d π β‘ π ] a d π π β π€ [ a d π , a d π ] = a d [ π , π ]
Proof of 1β3
Let
. Assuming anticommutativity π , π , π β π€ a d π β‘ [ π , π ] = [ π , [ π , π ] ] = β [ π , [ π , π ] ] β [ π , [ π , π ] ] = [ π , [ π , π ] ] + [ [ π , π ] , π ] = [ π , a d π β‘ π ] + [ a d π β‘ π , π ] proving ^P1.
Assume
. Then a d π β‘ ( π ) = [ π , π ] = π π 0 = [ π , [ π , π ] ] + [ π , [ π , π ] ] + [ π , [ π , π ] ] = [ π , π π ] + [ π , β π π ] = β π [ π , π ] = β π 2 π hence either
or π = 0 , proving ^P2. π = 0 For any
, π β π€ [ a d π , a d π ] ( π ) = a d π β‘ a d π β‘ ( π ) β a d π β‘ a d π β‘ ( π ) = [ π , [ π , π ] ] β [ π , [ π , π ] ] = [ π , [ π , π ] ] + [ π , [ π , π ] ] = β [ π , [ π , π ] ] = [ [ π , π ] , π ] = a d [ π , π ] β‘ ( π ) hence
, proving ^P3. [ a d π , a d π ] = a d [ π , π ]
Further terminology
- We say
π₯ β π€ a d a d π₯