Adjoint representation of a Lie algebra

Summary

The adjoint representation of a Lie algebra $\mathfrak g$ is the map $\operatorname{ad}$ that sends $x$ to the linear map $y \mapsto [x, y]$. By the Jacobi identity it is a Lie algebra homomorphism from $\mathfrak g$ to $\mathfrak{gl}(\mathfrak g)$, so $\mathfrak g$ acts on itself. Its kernel is the center of $\mathfrak g$. It is the derivative of the adjoint representation of a Lie group, and it underlies the Killing form and the root decomposition.