Definition

The adjoint representation of a Lie algebra (of a Lie group) is the map

where

differential of the adjoint representation of Lie group.

Intuition

We first start with the adjoint representation of Lie group

Then, we can take the differential

We can recognize . Looking closer at , we see that

Then we see that .

There are a couple important details here:

  • is a Lie group itself, so is the Lie algebra of , .

  • is a vector space, and the tangent space of a vector space can be identified with the vector space itself, so .

Based on the bullet points, this new map is Lie algebra representation since it is a Lie algebra homomorphism and we denote it as . Therefore we have the map,