Adjoint Representation of a Lie Algebra

Let $\mathfrak{g}$ be a Lie algebra with Lie bracket $[\ ,\ ]$ . For $X\in\mathfrak{g}$ , define a linear map

\operatorname{ad}_X:\mathfrak{g}\to \mathfrak{g},\qquad \operatorname{ad}_X(Y)=[X,Y].

This gives a map

\operatorname{ad}:\mathfrak{g}\to \mathfrak{gl}(\mathfrak{g}),\qquad X\mapsto \operatorname{ad}_X,

called the adjoint representation of $\mathfrak{g}$ .

Key property

The map $\operatorname{ad}$ is a Lie algebra homomorphism:

[\operatorname{ad}_X,\operatorname{ad}_Y]=\operatorname{ad}_{[X,Y]} \quad\text{in}\quad \mathfrak{gl}(\mathfrak{g}).

Thus $\operatorname{ad}$ is a representation of a Lie algebra on the vector space $\mathfrak{g}$ .

Kernel and center

$\ker(\operatorname{ad})$ consists of elements commuting with everything, i.e. the center $Z(\mathfrak{g})$ .

Killing form

The Killing form is defined by

B(X,Y)=\operatorname{tr}(\operatorname{ad}_X\operatorname{ad}_Y),

using the trace. It is fundamental for semisimple Lie algebras.

For a Lie group version, see adjoint action of a Lie group.