# Adjoint action of Lie group on Lie algebra

## Definition

### Definition for a real Lie group

Suppose $G$ is a real Lie group and $\mathfrak{g}$ is its Lie algebra. The adjoint action of $G$ on $\mathfrak{g}$ is a homomorphism of groups from $G$ to the automorphism group of $\mathfrak{g}$, i.e., a homomorphism:

$\operatorname{Ad}: G \to \operatorname{Aut}\mathfrak{g}$

The map is defined as follows: for $g \in G$, $\operatorname{Ad}(g)$ evaluated at $x \in \mathfrak{g}$ is defined as follows.

1. First, find the unique one-parameter group $\gamma:\R \to G$ such that $\gamma'(0) = x$.
2. Consider the new one-parameter group $\beta = c_g \circ \gamma$ where $c_g$ is the inner automorphism defined as conjugation by $g$. In other words, we define $\beta(t) = g\gamma(t)g^{-1}$.
3. Now, take the tangent vector $\beta'(0)$. This is the desired answer.

### Definition for Lie groups over other topological fields

This is similar to the definition for a real Lie group.

### Definition for a linear Lie group

Suppose $G$ is a linear Lie group over a topological field $K$, i.e., a Lie group with an embedding as a closed subgroup of the general linear group $GL(n,K)$ (where the closed is relative to the topology). Suppose $\mathfrak{g}$ is the Lie algebra of $G$. The adjoint action of $G$ on $\mathfrak{g}$ is a homomorphism of groups from $G$ to the automorphism group of $\mathfrak{g}$, i.e., a homomorphism:

$\operatorname{Ad}: G \to \operatorname{Aut}\mathfrak{g}$

The map is defined as follows: for $g \in G$ and $x \in \mathfrak{g}$, we define

$\operatorname{Ad}(g)(x) := gxg^{-1}$

where the multiplication on the right side is matrix multiplication.

### Definition for a Lazard Lie group

If $G$ is a Lazard Lie group, its Lazard Lie ring can be identified with $G$ as a set, with the Lie ring operations defined in terms of the group operations of $G$. The adjoint action of the group $G$ on itself as a Lie ring is simply the group action on itself by conjugation, now viewed as an action on itself as a Lie ring.