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 $\gamma :\mathbb {R} \to G$ 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.