Adjoint action of Lie group on Lie algebra

Definition

Definition for a real Lie group

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

$A d : G \to A u t g$

The map is defined as follows: for $g \in G$ , $A d (g)$ evaluated at $x \in g$ is defined as follows.

First, find the unique one-parameter group $γ : R \to G$ such that $γ^{'} (0) = x$ .
Consider the new one-parameter group $β = c_{g} \circ γ$ where $c_{g}$ is the inner automorphism defined as conjugation by $g$ . In other words, we define $β (t) = g γ (t) g^{- 1}$ .
Now, take the tangent vector $β^{'} (0)$ . 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 $G L (n, K)$ (where the closed is relative to the topology). Suppose $g$ is the Lie algebra of $G$ . The adjoint action of $G$ on $g$ is a homomorphism of groups from $G$ to the automorphism group of $g$ , i.e., a homomorphism:

$A d : G \to A u t g$

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

$A d (g) (x) : = g x g^{- 1}$

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.