Contents

Idea

An adjoint action is an action by conjugation .

Definition

Of a group on itself

The adjoint action of a group $G$ on itself is the action $\mathrm{Ad}:G\times G\to G$ given by

Of a Lie group on its Lie algebra

The adjoint action $\mathrm{ad}:G\times 𝔤\to 𝔤$ of a Lie group $G$ on its Lie algebra $𝔤$ is for each $g\in G$ the derivative $d\mathrm{Ad}(g):T_{e}G\to T_{e}G$ of this action in the second argument at the neutral element of $G$

This is often written as $\mathrm{ad}(g)(x)=g^{-1}xg$ even though for a general Lie group the expression on the right is not the product of three factors in any way. But for a matrix Lie group $G$ it is: in this case both $g$ as well as $x$ are canonically identified with matrices and the expression on the right is the product of these matrices.

Since this is a linear action, it is called the adjoint representation of a Lie group. The associated bundles with respect to this representation are called adjoint bundles.

Of a Lie algebra on itself

Differentiating the above example also in the second argument, yields the adjoint action of a Lie algebra on itself

which is simply the Lie bracket

Of a Hopf algebra on itself

Let $k$ be a commutative unital ring and $H=(H,m,\eta ,\Delta ,ϵ,S)$ be a Hopf $k$-algebra with multiplication $m$, unit map $\eta$, comultiplication $\Delta$, counit $ϵ$ and the antipode map $S:H\to H^{\mathrm{op}}$. We can use Sweedler notation $\Delta (h)=\sum h_{(1)}{\otimes }_k{h}_{(2)}$. The adjoint action of $H$ on $H$ is given by

and it makes $H$ not only an $H$-module, but in fact a monoid in the monoidal category of $H$-modules (usually called $H$-module algebra).

Related concepts

• coadjoint action

• adjoint representation, adjoint bundle

• conjugation action

• conjugacy class