adjoint representation


ad \DeclareMathOperator\EndEnd

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


to be the linear transformation with action


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


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


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


while the right hand side maps c to


Taking skew-symmetry of the bracket as a given, the equality of these two expressions is logically equivalent to the Jacobi identity:

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