adjoint representation


\DeclareMathOperator\ad

ad \DeclareMathOperator\EndEnd

Let \mathfrakg be a Lie algebraMathworldPlanetmath. For every a\mathfrakg we define the , a.k.a. the adjoint action,

\ad(a):\mathfrakg\mathfrakg

to be the linear transformation with action

\ad(a):b[a,b],b\mathfrakg.

For any vector spaceMathworldPlanetmath V, we use \mathfrakgl(V) to denote the Lie algebra of \EndV determined by the commutator bracket. So \mathfrakgl(V)=\EndV as vector spaces, only the multiplications are different.

In this notation, treating \mathfrakg as a vector space, the linear mapping \ad:\mathfrakg\mathfrakgl(\mathfrakg) with action

a\ad(a),a\mathfrakg

is called the adjoint representation of \mathfrakg. The fact that \ad defines a representationPlanetmathPlanetmath is a straight-forward consequence of the Jacobi identityMathworldPlanetmath axiom. Indeed, let a,b\mathfrakg be given. We wish to show that

\ad([a,b])=[\ad(a),\ad(b)],

where the bracket on the left is the \mathfrakg multiplication structureMathworldPlanetmath, and the bracket on the right is the commutator bracket. For all c\mathfrakg 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