adjoint representation

ad End

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 with action

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

For any vector space $V$, we use $gl(V)$ to denote the Lie algebra of $V$ 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 of the Jacobi identity axiom. Indeed, let $a,b\in g$ be given. We wish 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 the commutator 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 a given, the equality of these two expressions is logically equivalent to the Jacobi identity:

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

Title	adjoint representation
Canonical name	AdjointRepresentation
Date of creation	2015-10-05 17:38:19
Last modified on	2015-10-05 17:38:19
Owner	rmilson (146)
Last modified by	rmilson (146)
Numerical id	9
Author	rmilson (146)
Entry type	Definition
Classification	msc 17B10
Related topic	IsotropyRepresentation
Defines	adjoint action
Defines	gl
Defines	general linear Lie algebra