Adjoint action of Lie group on Lie algebra

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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.