Reference request (or otherwise): Adjoint action

Question

I am interested in clearing up a confusion of mine. I will try to make my question as clear as I can but I apologize in advance if this is not the case.

Given a unitary group of some unital involutive algebra, $U(A) =$ {$u \in A| uu^* = u^*u = I $},

I can define the adjoint action (and right action shortly) of u as $(Ad u)\xi:=u\xi u^*$, Where $\xi \in H$, the Hilbert space on which I am representing the algebra $A$.

Define right action on elements of the hilbert space using Connes Real Structure operator J. $\xi u^* = JuJ^*\xi$. At the level of the algebra the adjoint is then given as $(ad B)\xi:= B\xi - \xi B$.

My Question: How does this generalize in the case where the algebra A is non-associative and H is a general 'bimodule' over A? is conjugation always well defined?

For example. Say I am interested in the group G2 which is generated by the derivation algebra of the octonions (ie it is the automorphism group of the octonions). I can write out the root diagram for G2 so I know there should be an adjoint representation. The octonions O however are are not-associative. As further example if I take A = H = O, how is the adjoint action and conjugation defined (explicitly)?

I imagine there is a very simple answer to this question. Any direct help you could offer or an appropriate reference would be greatly appreciated.

Answer 1

I hope this clears up some of your confusion:

1. The group $U(A)$ acts on $A$ itself by conjugation: $\mathrm{Ad}_u(a) = uau^\ast$.

2. If $A$ acts on a Hilbert space $H$ then the group $U(A)$ acts on $H$ also just by restricting the action of $A$ on $H$. What you have written, $\mathrm{Ad}_u(\xi) = u \xi u^\ast$, doesn't make sense. What does it mean to multiply a vector on the right by an operator?

3. What is true is that there is a certain intertwining relation between the conjugation action of $U(A)$ on $A$ and the action of $U(A)$ on $H$. Specifically: $$ u (a \xi) = u a u^\ast u \xi = \mathrm{Ad}_u(a)(u \xi) $$

4. The other adjoint action that you have written, $\mathrm{ad}_B(\xi) = B\xi - \xi B$, also does not make sense as stated, for the same reason as above: it doesn't make sense to multiply a vector on the right by an operator. This type of adjoint action is usually associated with Lie algebras. Since $A$ is an associative algebra, it is a Lie algebra with the bracket $[a,b] = ab - ba$. For any Lie algebra, the adjoint action is defined as $\mathrm{ad}_X(Y) = [X,Y]$.

5. For Lie groups and Lie algebras, the relation between these two notions of adjoint is standard, and not really at the appropriate level for this site. But it works like this: let $G$ be a Lie group with Lie algebra $\mathfrak{g}$. For $g \in G$, define $C_g : G \to G$ by $C_g(x) = gxg^{-1}$. This is a smooth map which takes the identity to the identity. The differential of this map at the identity is then a linear map from $\mathfrak{g}$ to $\mathfrak{g}$, which is called $\mathrm{Ad}_g$. Then the map $Ad : g \mapsto Ad_g$ is a smooth group homomorphism from $G$ to $GL(\mathfrak{g})$; the differential of this map at the identity is a linear map $ad : \mathfrak{g} \to \mathfrak{gl(g)}$. You can check that this agrees with the notion of adjoint defined in item 4 above.

I would suggest thinking about these different notions of adjoint for a while and then coming back to ask a more focused question.