# Adjoint action of Lie group on Lie algebra

## Contents

## Definition

### Definition for a real Lie group

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

The map is defined as follows: for , evaluated at is defined as follows.

- First, find the unique one-parameter group such that .
- Consider the new one-parameter group where is the inner automorphism defined as conjugation by . In other words, we define .
- Now, take the tangent vector . 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 is a linear Lie group over a topological field , i.e., a Lie group with an embedding as a closed subgroup of the general linear group (where the *closed* is relative to the topology). Suppose is the Lie algebra of . The **adjoint action** of on is a homomorphism of groups from to the automorphism group of , i.e., a homomorphism:

The map is defined as follows: for and , we define

where the multiplication on the right side is matrix multiplication.

### Definition for a Lazard Lie group

If is a Lazard Lie group, its Lazard Lie ring can be identified with as a set, with the Lie ring operations defined in terms of the group operations of . The adjoint action of the group 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.