adjoint representation

adEnd

Let $g$ be a Lie algebra. For every $a\in g$ we define the, a.k.a. the adjoint action,

$$(a):g\to g$$

to be the linear transformation withaction

$$(a):b\mapsto [a,b],b\in g.$$

For any vector space $V$, we use $gl(V)$ to denote the Lie algebraof $V$ determined by the commutator bracket. So$gl(V)=V$ as vector spaces, only the multiplications are different.

In this notation, treating $g$ as a vector space, the linear mapping $:g\to gl(g)$ with action

$$a\mapsto (a),a\in g$$

is called the adjoint representation of $g$. The fact that$$ defines a representation is a straight-forward consequence ofthe Jacobi identity axiom. Indeed, let $a,b\in g$ be given. Wewish to show that

$$([a,b])=[(a),(b)],$$

where the bracket on the left is the$g$ multiplication structure, and the bracket on the right is thecommutator bracket. For all $c\in g$ the left hand side maps $c$ to

$$[[a,b],c],$$

while the right hand side maps $c$ to

$$[a,[b,c]]+[b,[a,c]].$$

Taking skew-symmetry of the bracket as agiven, the equality of these two expressions is logically equivalentto the Jacobi identity:

$$[a,[b,c]]+[b,[c,a]]+[c,[a,b]]=0.$$