### Context

#### Algebra

higher algebra

universal algebra

# 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 $\mathrm{Ad}:G×G\to G$ given by

$\mathrm{Ad}:\left(g,h\right)↦{g}^{-1}\cdot h\cdot g\phantom{\rule{thinmathspace}{0ex}}.$Ad : (g,h) \mapsto g^{-1} \cdot h \cdot g \,.

### Of a Lie group on its Lie algebra

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

$\mathrm{ad}:\left(g,x\right)↦\mathrm{Ad}\left(g{\right)}_{*}\left(x\right)\phantom{\rule{thinmathspace}{0ex}}.$ad : (g,x) \mapsto Ad(g)_*(x) \,.

This is often written as $\mathrm{ad}\left(g\right)\left(x\right)={g}^{-1}xg$ 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

$\mathrm{ad}:𝔤×𝔤\to 𝔤$ad : \mathfrak{g} \times \mathfrak{g} \to \mathfrak{g}

which is simply the Lie bracket

${\mathrm{ad}}_{x}:y↦\left[x,y\right]\phantom{\rule{thinmathspace}{0ex}}.$ad_x : y \mapsto [x,y] \,.

### Of a Hopf algebra on itself

Let $k$ be a commutative unital ring and $H=\left(H,m,\eta ,\Delta ,ϵ,S\right)$ 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 $\Delta \left(h\right)=\sum {h}_{\left(1\right)}{\otimes }_{k}{h}_{\left(2\right)}$. The adjoint action of $H$ on $H$ is given by

$h▹g=\sum {h}_{\left(1\right)}gS\left({h}_{\left(2\right)}\right)$h\triangleright g = \sum h_{(1)} g S(h_{(2)})

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).

Revised on February 8, 2013 12:21:55 by Urs Schreiber (89.204.138.214)